From f0e571cdbcef9f983ec0b598948f48cc422ee577 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Sun, 2 Jun 2024 16:41:05 +0000 Subject: [PATCH] build based on c7a4c23 --- previews/PR604/.documenter-siteinfo.json | 1 + previews/PR604/api/index.html | 84 ++ previews/PR604/assets/documenter.js | 1050 +++++++++++++++++ .../PR604/assets/themes/documenter-dark.css | 7 + .../PR604/assets/themes/documenter-light.css | 9 + previews/PR604/assets/themeswap.js | 84 ++ previews/PR604/assets/warner.js | 52 + previews/PR604/index.html | 17 + previews/PR604/manual/codegen/index.html | 36 + previews/PR604/manual/interface/index.html | 2 + .../PR604/manual/representation/index.html | 26 + previews/PR604/manual/rewrite/index.html | 50 + previews/PR604/objects.inv | Bin 0 -> 1072 bytes previews/PR604/search_index.js | 3 + previews/PR604/siteinfo.js | 1 + previews/PR604/upgrade/index.html | 2 + 16 files changed, 1424 insertions(+) create mode 100644 previews/PR604/.documenter-siteinfo.json create mode 100644 previews/PR604/api/index.html create mode 100644 previews/PR604/assets/documenter.js create mode 100644 previews/PR604/assets/themes/documenter-dark.css create mode 100644 previews/PR604/assets/themes/documenter-light.css create mode 100644 previews/PR604/assets/themeswap.js create mode 100644 previews/PR604/assets/warner.js create mode 100644 previews/PR604/index.html create mode 100644 previews/PR604/manual/codegen/index.html create mode 100644 previews/PR604/manual/interface/index.html create mode 100644 previews/PR604/manual/representation/index.html create mode 100644 previews/PR604/manual/rewrite/index.html create mode 100644 previews/PR604/objects.inv create mode 100644 previews/PR604/search_index.js create mode 100644 previews/PR604/siteinfo.js create mode 100644 previews/PR604/upgrade/index.html diff --git a/previews/PR604/.documenter-siteinfo.json b/previews/PR604/.documenter-siteinfo.json new file mode 100644 index 000000000..f6577f4ad --- /dev/null +++ b/previews/PR604/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-06-02T16:41:02","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/previews/PR604/api/index.html b/previews/PR604/api/index.html new file mode 100644 index 000000000..1152ee26c --- /dev/null +++ b/previews/PR604/api/index.html @@ -0,0 +1,84 @@ + +API Reference · SymbolicUtils.jl
GitHub

API Reference

Symbols and Terms

SymbolicUtils.@syms — Macro
@syms <lhs_expr>[::T1] <lhs_expr>[::T2]...

For instance:

@syms foo::Real bar baz(x, y::Real)::Complex

Create one or more variables. <lhs_expr> can be just a symbol in which case it will be the name of the variable, or a function call in which case a function-like variable which has the same name as the function being called. The Sym type, or in the case of a function-like Sym, the output type of calling the function can be set using the ::T syntax.

Examples:

  • @syms foo bar::Real baz::Int will create

variable foo of symtype Number (the default), bar of symtype Real and baz of symtype Int

  • @syms f(x) g(y::Real, x)::Int h(a::Int, f(b)) creates 1-arg f 2-arg g

and 2 arg h. The second argument to h must be a one argument function-like variable. So, h(1, g) will fail and h(1, f) will work.

source
Missing docstring.

Missing docstring for SymbolicUtils.Sym. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SymbolicUtils.symtype. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SymbolicUtils.Term. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SymbolicUtils.Add. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SymbolicUtils.Mul. Check Documenter's build log for details.

Missing docstring.

Missing docstring for SymbolicUtils.Pow. Check Documenter's build log for details.

SymbolicUtils.promote_symtype — Function
promote_symtype(f, Ts...)

The result of applying f to arguments of symtype Ts...

julia> promote_symtype(+, Real, Real)
+Real
+
+julia> promote_symtype(+, Complex, Real)
+Number
+
+julia> @syms f(x)::Complex
+(f(::Number)::Complex,)
+
+julia> promote_symtype(f, Number)
+Complex

When constructing Terms without an explicit symtype, promote_symtype is used to figure out the symtype of the Term.

source
promote_symtype(f::FnType{X,Y}, arg_symtypes...)

The output symtype of applying variable f to arguments of symtype arg_symtypes.... if the arguments are of the wrong type then this function will error.

source

Rewriters

SymbolicUtils.@rule — Macro
@rule LHS => RHS

Creates a Rule object. A rule object is callable, and takes an expression and rewrites it if it matches the LHS pattern to the RHS pattern, returns nothing otherwise. The rule language is described below.

LHS can be any possibly nested function call expression where any of the arguments can optionally be a Slot (~x) or a Segment (~~x) (described below).

If an expression matches LHS entirely, then it is rewritten to the pattern in the RHS Segment (~x) and slot variables (~~x) on the RHS will substitute the result of the matches found for these variables in the LHS.

Slot:

A Slot variable is written as ~x and matches a single expression. x is the name of the variable. If a slot appears more than once in an LHS expression then expression matched at every such location must be equal (as shown by isequal).

Example:

Simple rule to turn any sin into cos:

julia> @syms a b c
+(a, b, c)
+
+julia> r = @rule sin(~x) => cos(~x)
+sin(~x) => cos(~x)
+
+julia> r(sin(1+a))
+cos((1 + a))

A rule with 2 segment variables

julia> r = @rule sin(~x + ~y) => sin(~x)*cos(~y) + cos(~x)*sin(~y)
+sin(~x + ~y) => sin(~x) * cos(~y) + cos(~x) * sin(~y)
+
+julia> r(sin(a + b))
+cos(a)*sin(b) + sin(a)*cos(b)

A rule that matches two of the same expressions:

julia> r = @rule sin(~x)^2 + cos(~x)^2 => 1
+sin(~x) ^ 2 + cos(~x) ^ 2 => 1
+
+julia> r(sin(2a)^2 + cos(2a)^2)
+1
+
+julia> r(sin(2a)^2 + cos(a)^2)
+# nothing

Segment:

A Segment variable is written as ~~x and matches zero or more expressions in the function call.

Example:

This implements the distributive property of multiplication: +(~~ys) matches expressions like a + b, a+b+c and so on. On the RHS ~~ys presents as any old julia array.

julia> r = @rule ~x * +((~~ys)) => sum(map(y-> ~x * y, ~~ys));
+
+julia> r(2 * (a+b+c))
+((2 * a) + (2 * b) + (2 * c))

Predicates:

There are two kinds of predicates, namely over slot variables and over the whole rule. For the former, predicates can be used on both ~x and ~~x by using the ~x::f or ~~x::f. Here f can be any julia function. In the case of a slot the function gets a single matched subexpression, in the case of segment, it gets an array of matched expressions.

The predicate should return true if the current match is acceptable, and false otherwise.

julia> two_πs(x::Number) = abs(round(x/(2π)) - x/(2π)) < 10^-9
+two_πs (generic function with 1 method)
+
+julia> two_πs(x) = false
+two_πs (generic function with 2 methods)
+
+julia> r = @rule sin(~~x + ~y::two_πs + ~~z) => sin(+(~~x..., ~~z...))
+sin(~(~x) + ~(y::two_πs) + ~(~z)) => sin(+(~(~x)..., ~(~z)...))
+
+julia> r(sin(a+3π))
+
+julia> r(sin(a+6π))
+sin(a)
+
+julia> r(sin(a+6π+c))
+sin((a + c))

Predicate function gets an array of values if attached to a segment variable (~~x).

For the predicate over the whole rule, use @rule <LHS> => <RHS> where <predicate>:

julia> @syms a b;
+
+julia> predicate(x) = x === a;
+
+julia> r = @rule ~x => ~x where f(~x);
+
+julia> r(a)
+a
+
+julia> r(b) === nothing
+true

Note that this is syntactic sugar and that it is the same as something like @rule ~x => f(~x) ? ~x : nothing.

Context:

In predicates: Contextual predicates are functions wrapped in the Contextual type. The function is called with 2 arguments: the expression and a context object passed during a call to the Rule object (maybe done by passing a context to simplify or a RuleSet object).

The function can use the inputs however it wants, and must return a boolean indicating whether the predicate holds or not.

In the consequent pattern: Use (@ctx) to access the context object on the right hand side of an expression.

source
SymbolicUtils.Rewriters — Module

A rewriter is any function which takes an expression and returns an expression or nothing. If nothing is returned that means there was no changes applicable to the input expression.

The Rewriters module contains some types which create and transform rewriters.

  • Empty() is a rewriter which always returns nothing
  • Chain(itr) chain an iterator of rewriters into a single rewriter which applies each chained rewriter in the given order. If a rewriter returns nothing this is treated as a no-change.
  • RestartedChain(itr) like Chain(itr) but restarts from the first rewriter once on the first successful application of one of the chained rewriters.
  • IfElse(cond, rw1, rw2) runs the cond function on the input, applies rw1 if cond returns true, rw2 if it returns false
  • If(cond, rw) is the same as IfElse(cond, rw, Empty())
  • Prewalk(rw; threaded=false, thread_cutoff=100) returns a rewriter which does a pre-order traversal of a given expression and applies the rewriter rw. Note that if rw returns nothing when a match is not found, then Prewalk(rw) will also return nothing unless a match is found at every level of the walk. threaded=true will use multi threading for traversal. thread_cutoff is the minimum number of nodes in a subtree which should be walked in a threaded spawn.
  • Postwalk(rw; threaded=false, thread_cutoff=100) similarly does post-order traversal.
  • Fixpoint(rw) returns a rewriter which applies rw repeatedly until there are no changes to be made.
  • FixpointNoCycle behaves like Fixpoint but instead it applies rw repeatedly only while it is returning new results.
  • PassThrough(rw) returns a rewriter which if rw(x) returns nothing will instead return x otherwise will return rw(x).
source

Simplify

SymbolicUtils.simplify — Function
simplify(x; expand=false,
+            threaded=false,
+            thread_subtree_cutoff=100,
+            rewriter=nothing)

Simplify an expression (x) by applying rewriter until there are no changes. expand=true applies expand in the beginning of each fixpoint iteration.

By default, simplify will assume denominators are not zero and allow cancellation in fractions. Pass simplify_fractions=false to prevent this.

source
SymbolicUtils.expand — Function
expand(expr)

Expand expressions by distributing multiplication over addition, e.g., a*(b+c) becomes ab+ac.

expand uses replace symbols and non-algebraic expressions by variables of type variable_type to compute the distribution using a specialized sparse multivariate polynomials implementation. variable_type can be any subtype of MultivariatePolynomials.AbstractVariable.

source
SymbolicUtils.substitute — Function
substitute(expr, dict; fold=true)

substitute any subexpression that matches a key in dict with the corresponding value. If fold=false, expressions which can be evaluated won't be evaluated.

julia> substitute(1+sqrt(y), Dict(y => 2), fold=true)
+2.414213562373095
+julia> substitute(1+sqrt(y), Dict(y => 2), fold=false)
+1 + sqrt(2)
source

Utilities

SymbolicUtils.@timerewrite — Macro
@timerewrite expr

If expr calls simplify or a RuleSet object, track the amount of time it spent on applying each rule and pretty print the timing.

This uses TimerOutputs.jl.

Example:


+julia> expr = foldr(*, rand([a,b,c,d], 100))
+(a ^ 26) * (b ^ 30) * (c ^ 16) * (d ^ 28)
+
+julia> @timerewrite simplify(expr)
+ ────────────────────────────────────────────────────────────────────────────────────────────────
+                                                         Time                   Allocations
+                                                 ──────────────────────   ───────────────────────
+                Tot / % measured:                     340ms / 15.3%           92.2MiB / 10.8%
+
+ Section                                 ncalls     time   %tot     avg     alloc   %tot      avg
+ ────────────────────────────────────────────────────────────────────────────────────────────────
+ ACRule((~y) ^ ~n * ~y => (~y) ^ (~n ...    667   11.1ms  21.3%  16.7μs   2.66MiB  26.8%  4.08KiB
+   RHS                                       92    277μs  0.53%  3.01μs   14.4KiB  0.14%     160B
+ ACRule((~x) ^ ~n * (~x) ^ ~m => (~x)...    575   7.63ms  14.6%  13.3μs   1.83MiB  18.4%  3.26KiB
+ (*)(~(~(x::!issortedₑ))) => sort_arg...    831   6.31ms  12.1%  7.59μs    738KiB  7.26%     910B
+   RHS                                      164   3.03ms  5.81%  18.5μs    250KiB  2.46%  1.52KiB
+   ...
+   ...
+ ────────────────────────────────────────────────────────────────────────────────────────────────
+(a ^ 26) * (b ^ 30) * (c ^ 16) * (d ^ 28)
source
diff --git a/previews/PR604/assets/documenter.js b/previews/PR604/assets/documenter.js new file mode 100644 index 000000000..c6562b558 --- /dev/null +++ b/previews/PR604/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR604/assets/themes/documenter-dark.css b/previews/PR604/assets/themes/documenter-dark.css new file mode 100644 index 000000000..1d7170158 --- /dev/null +++ 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.pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:active,html.theme--documenter-dark .button:active,html.theme--documenter-dark .is-active.pagination-previous,html.theme--documenter-dark .is-active.pagination-next,html.theme--documenter-dark .is-active.pagination-link,html.theme--documenter-dark .is-active.pagination-ellipsis,html.theme--documenter-dark .is-active.file-cta,html.theme--documenter-dark .is-active.file-name,html.theme--documenter-dark .select select.is-active,html.theme--documenter-dark .is-active.textarea,html.theme--documenter-dark .is-active.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-active,html.theme--documenter-dark .is-active.button{outline:none}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled],html.theme--documenter-dark .pagination-ellipsis[disabled],html.theme--documenter-dark .file-cta[disabled],html.theme--documenter-dark .file-name[disabled],html.theme--documenter-dark .select select[disabled],html.theme--documenter-dark .textarea[disabled],html.theme--documenter-dark .input[disabled],html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input[disabled],html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark fieldset[disabled] .pagination-previous,fieldset[disabled] html.theme--documenter-dark .pagination-next,html.theme--documenter-dark fieldset[disabled] .pagination-next,fieldset[disabled] html.theme--documenter-dark .pagination-link,html.theme--documenter-dark fieldset[disabled] .pagination-link,fieldset[disabled] html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark fieldset[disabled] .pagination-ellipsis,fieldset[disabled] html.theme--documenter-dark .file-cta,html.theme--documenter-dark fieldset[disabled] .file-cta,fieldset[disabled] html.theme--documenter-dark .file-name,html.theme--documenter-dark fieldset[disabled] .file-name,fieldset[disabled] html.theme--documenter-dark .select select,fieldset[disabled] html.theme--documenter-dark .textarea,fieldset[disabled] html.theme--documenter-dark .input,fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar 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.button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--documenter-dark .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-white.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus:not(:active),html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:focus:not(:active),html.theme--documenter-dark .button.is-primary.is-focused:not(:active),html.theme--documenter-dark .docstring>section>a.button.is-focused.docs-sourcelink:not(:active){box-shadow:0 0 0 0.125em rgba(55,90,127,0.25)}html.theme--documenter-dark .button.is-primary:active,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:active,html.theme--documenter-dark .button.is-primary.is-active,html.theme--documenter-dark .docstring>section>a.button.is-active.docs-sourcelink{background-color:#2f4d6d;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary[disabled],html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-primary,fieldset[disabled] html.theme--documenter-dark 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ad8100 #ad8100 !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fffaeb;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fff7de;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fff3d1;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:transparent;color:#fff}html.theme--documenter-dark 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.button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--documenter-dark .icon.is-medium{height:2rem;width:2rem}html.theme--documenter-dark .icon.is-large{height:3rem;width:3rem}html.theme--documenter-dark .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--documenter-dark .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--documenter-dark .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--documenter-dark .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--documenter-dark div.icon-text{display:flex}html.theme--documenter-dark .image,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--documenter-dark .image img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--documenter-dark .image.is-4by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--documenter-dark .image.is-4by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--documenter-dark .image.is-3by2 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--documenter-dark .image.is-3by2 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--documenter-dark .image.is-5by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--documenter-dark .image.is-5by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--documenter-dark .image.is-16by9 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/previews/PR604/assets/themeswap.js b/previews/PR604/assets/themeswap.js new file mode 100644 index 000000000..9f5eebe6a --- /dev/null +++ b/previews/PR604/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/previews/PR604/assets/warner.js b/previews/PR604/assets/warner.js new file mode 100644 index 000000000..3f6f5d008 --- /dev/null +++ b/previews/PR604/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/previews/PR604/index.html b/previews/PR604/index.html new file mode 100644 index 000000000..7256eb667 --- /dev/null +++ b/previews/PR604/index.html @@ -0,0 +1,17 @@ + +SymbolicUtils.jl — Symbolic programming in Julia · SymbolicUtils.jl
GitHub

SymbolicUtils.jl — Symbolic programming in Julia

Features

Creating symbolic expressions

First, let's use the @syms macro to create a few symbols.

using SymbolicUtils
+
+@syms w z α::Real β::Real
+
+(w, z, α, β) # hide
+

Type annotations are optional when creating symbols. Here α, β behave like Real numbers. w and z behave like Number, which is the default. You can use the symtype function to find the type of a symbol.

using SymbolicUtils: symtype
+
+symtype(w), symtype(z),  symtype(α), symtype(β)

Note however that they are not subtypes of these types!

@show w isa Number
+@show α isa Real

As their types are different:

@show typeof(w)
+@show typeof(α)

(see this post for why they are all not just subtypes of Number)

You can do basic arithmetic on symbols to get symbolic expressions:

expr1 = α*sin(w)^2 + β*cos(z)^2
+expr2 = α*cos(z)^2 + β*sin(w)^2
+
+expr1 + expr2

SymbolicUtils automatically simplifies

2w + 3w - 3z + α

and reorders

(z + w)*(α + β)

expressions of type Symbolic{<:Number} (which includes Sym{Real}) when they are created. It also does constant elimination (including rational numbers)

5 + 2w - 3z + α - (β + 5//3) + 3w - 2 + 3//2 * β

It's worth remembering that the expression may be transformed with respect to the input when it's created.

Function-like symbols

Symbols can be defined to behave like functions. Both the input and output types for the function can be specified. Any application to that function will only admit either values of those types or symbols of the same symtype.

using SymbolicUtils
+@syms f(x) g(x::Real, y::Real)::Real
+
+f(z) + g(1, α) + sin(w)

This does not work since z is a Number, not a Real.

g(1, z)

This works because g "returns" a Real.

g(2//5, g(1, β))

Expression interface

Symbolic expressions are of type Term{T}, Add{T}, Mul{T}, Pow{T} or Div{T} and denote some function call where one or more arguments are themselves such expressions or Syms. See more about the representation here.

All the expression types support the TermInterface.jl interface. Please refer to the package for the complete reference of the interface.

Term rewriting

SymbolicUtils contains a rule-based rewriting language for easy pattern matching and rewriting of expression. There is also a combinator library to combine rules to chain, branch and loop over rules.

Simplification

By default +, * and ^ operations apply the most basic simplification upon construction of the expression.

The rules with which the canonical form of Symbolic{<:Number} terms are constructed are the next (where x isa Symbolic and c isa Number)

  • 0 + x, 1 * x and x^1 always gives x
  • 0 * x always gives 0 and x ^ 0 gives 1
  • -x becomes (-1)*x
  • commutativity and associativity over + and * are assumed. Re-ordering of terms will be done under a total order
  • p/q * x or x * p/q results in (p*x)/q
  • p/q * x/y results in (p*x)/(q*y)
  • x + ... + x will be fused into n*x with type Mul
  • x * ... * x will be fused into x^n with type Pow
  • sum of Add's are fused
  • product of Mul's are fused
  • c * (c₁x₁ + ... + cₙxₙ) will be converted into c*c₁*x₁ + ... + c*cₙ*xₙ
  • (x₁^c₁ * ... * xₙ^cₙ)^c will be converted into x₁^(c*c₁) * ... * xₙ^(c*cₙ)
  • there are come other simplifications on construction that you can check here

Here is an example of this

2 * (w+w+α+β + sin(z)^2 + cos(z)^2 - 1)

The simplify function applies a built-in set of rules to rewrite expressions in order to simplify it.

simplify(2 * (w+w+α+β + sin(z)^2 + cos(z)^2 - 1))

The rules in the default simplify applies simple constant elimination and trigonometric identities.

If you read the previous section on the rules DSL, you should be able to read and understand the rules that are used by simplify.

Code generation

Experimental feature

It is common to want to generate executable code from symbolic expressions and blocks of them. We are working on experimental support for turning Symbolic expressions into executable functions with specific focus on array input and output and performance which is critical to the Differential Equations ecosystem which is making heavy use of this package.

See Code generation for more about this.

Learn more

If you have a package that you would like to utilize rule-based rewriting in, look at the suggestions in the Interfacing section to find out how you can do that without any fundamental changes to your package. Look at the API documentation for docstrings about specific functions or macros.

Head over to the github repository to ask questions and report problems! Join the Zulip stream to chat!

diff --git a/previews/PR604/manual/codegen/index.html b/previews/PR604/manual/codegen/index.html new file mode 100644 index 000000000..8568b14cc --- /dev/null +++ b/previews/PR604/manual/codegen/index.html @@ -0,0 +1,36 @@ + +Code generation · SymbolicUtils.jl
GitHub

Code generation

Note: this feature is experimental and the API might change frequently

toexpr(ex) turns any expression (ex) into the equivalent Expr which is suitable for eval. The SymbolicUtils.Code module provides some combinators which provides the ability to construct more complex expressions than just function calls. These include:

  • Let blocks
  • Functions with arguments and keyword arguments
    • Functions with arguments which are to be de-structured
  • Expressions that set array elements in-place
  • Expressions that create an array similar in type to a reference array (currently supports Array, StaticArrays.SArray, and LabelledArrays.SLArray)
  • Expressions that create sparse arrays

Do using SymbolicUtils.Code to get the following bindings

toexpr

SymbolicUtils.Code.toexpr — Function
toexpr(ex, [st,])

Convert a symbolic expression into an Expr, suitable to be passed into eval.

For example,

julia> @syms a b
+(a, b)
+
+julia> toexpr(a+b)
+:((+)(a, b))
+
+julia> toexpr(a+b) |> dump
+Expr
+  head: Symbol call
+  args: Array{Any}((3,))
+    1: + (function of type typeof(+))
+    2: Symbol a
+    3: Symbol b

Note that the function is an actual function object.

For more complex expressions, see other code-related combinators,

Namely Assignment, Let, Func, SetArray, MakeArray, MakeSparseArray and MakeTuple.

To make your own type convertible to Expr using toexpr define toexpr(x, st) and forward the state st in internal calls to toexpr. st is state used to know when to leave something like y(t) as it is or when to make it var"y(t)". E.g. when y(t) is itself the argument of a function rather than y.

source

Code Combinators

These are all exported when you do using SymbolicUtils.Code

SymbolicUtils.Code.Let — Type
Let(assignments, body[, let_block])

A Let block.

  • assignments is a vector of Assignments
  • body is the body of the let block
  • let_block boolean (default=true) – do not create a let block if false.
source
SymbolicUtils.Code.Func — Type
Func(args, kwargs, body[, pre])

A function.

  • args is a vector of expressions
  • kwargs is a vector of Assignments
  • body is the body of the function
  • pre a vector of expressions to be prepended to the function body, for example, it could be [Expr(:meta, :inline), Expr(:meta, :propagate_inbounds)] to create an @inline @propagate_inbounds function definition.

Special features in args:

  • args can contain DestructuredArgs
  • call expressions

For example,


+julia> @syms a b c t f(d) x(t) y(t) z(t)
+(a, b, c, t, f(::Number)::Number, x(::Number)::Number, y(::Number)::Number, z(::Number)::Number)
+
+julia> func = Func([a,x(t), DestructuredArgs([b, y(t)]), f], # args
+                   [c ← 2, z(t) ← 42], # kwargs
+                   f((a + b + c) / x(t) + y(t) + z(t)));
+
+julia> toexpr(func)
+:(function (a, var"x(t)", var"##arg#255", f; c = 2, var"z(t)" = 42)
+      let b = var"##arg#255"[1], var"y(t)" = var"##arg#255"[2]
+          f((+)(var"y(t)", var"z(t)", (*)((+)(a, b, c), (inv)(var"x(t)"))))
+      end
+  end)
  • the second argument is a DestructuredArgs, in the Expr form, it is given a random name, and is expected to receive a vector or tuple of size 2 containing the values of b and y(t). The let block that is automatically generated "destructures" these arguments.
  • x(t) and y(t) have been replaced with var"x(t)" and var"y(t)" symbols throughout

the generated Expr. This makes sure that we are not actually calling the expressions x(t) or y(t) but instead passing the right values in place of the whole expression.

  • f is also a function-like symbol, same as x and y, but since the args array contains f as itself rather than as say, f(t), it does not become a var"f(t)". The generated function expects a function of one argument to be passed in the position of f.

An example invocation of this function is:

julia> executable = eval(toexpr(func))
+#10 (generic function with 1 method)
+
+julia> executable(1, 2.0, [2,3.0], x->string(x); var"z(t)" = sqrt(42))
+"11.98074069840786"
source
SymbolicUtils.Code.SpawnFetch — Type
SpawnFetch{ParallelType}(funcs [, args], reduce)

Run every expression in funcs in its own task, the expression should be a Func object and is passed to Threads.Task(f). If Func takes arguments, then the arguments must be passed in as args–a vector of vector of arguments to each function in funcs. We don't use @spawn in order to support RuntimeGeneratedFunctions which disallow closures, instead we interpolate these functions or closures as smaller RuntimeGeneratedFunctions.

reduce function is used to combine the results of executing exprs. A SpawnFetch expression returns the reduced result.

Use Symbolics.MultithreadedForm ParallelType from the Symbolics.jl package to get the RuntimeGeneratedFunction version SpawnFetch.

ParallelType can be used to define more parallelism types SymbolicUtils supports Multithreaded type. Which spawns threaded tasks.

source
SymbolicUtils.Code.SetArray — Type
SetArray(inbounds, arr, elems)

An expression representing setting of elements of arr.

By default, every element of elems is copied over to arr,

but if elems contains AtIndex(i, val) objects, then arr[i] = val is performed in its place.

inbounds is a boolean flag, true surrounds the resulting expression in an @inbounds.

source
SymbolicUtils.Code.MakeArray — Type
MakeArray(elems, similarto, [output_eltype=nothing])

An expression which constructs an array.

  • elems is the output array
  • similarto can either be a type, or some symbol that is an array whose type needs to be emulated. If similarto is a StaticArrays.SArray, then the output array is also created as an SArray, similarly, an Array will result in an Array, and a LabelledArrays.SLArray will result in a labelled static array.
  • output_eltype: if set, then forces the element type of the output array to be this. by default, the output type is inferred automatically.

You can define:

@inline function create_array(A::Type{<:MyArray},a
+                              ::Nothing, d::Val{dims}, elems...) where dims
+
+# and
+
+@inline function create_array(::Type{<:MyArray}, T, ::Val{dims}, elems...) where dims

which creates an array of size dims using the elements elems and eltype T, to allow MakeArray to create arrays similarto MyArrays.

source
SymbolicUtils.Code.MakeSparseArray — Type
MakeSpaseArray(array)

An expression which creates a SparseMatrixCSC or a SparseVector.

The generated expression contains the sparsity information of array,

it only creates the nzval field at run time.

source
diff --git a/previews/PR604/manual/interface/index.html b/previews/PR604/manual/interface/index.html new file mode 100644 index 000000000..5e6054265 --- /dev/null +++ b/previews/PR604/manual/interface/index.html @@ -0,0 +1,2 @@ + +Interfacing with SymbolicUtils.jl · SymbolicUtils.jl
GitHub

Interfacing with SymbolicUtils.jl

This section is for Julia package developers who may want to use the simplify and rule rewriting system on their own expression types.

Defining the interface

SymbolicUtils matchers can match any Julia object that implements an interface to traverse it as a tree. The interface in question, is defined in the TermInterface.jl package. Its purpose is to provide a shared interface between various symbolic programming Julia packages.

In particular, you should define methods from TermInterface.jl for an expression tree type T with symbol types S to work with SymbolicUtils.jl

You can read the documentation of TermInterface.jl on the Github repository.

SymbolicUtils.jl only methods

promote_symtype(f, arg_symtypes...)

Returns the appropriate output type of applying f on arguments of type arg_symtypes.

diff --git a/previews/PR604/manual/representation/index.html b/previews/PR604/manual/representation/index.html new file mode 100644 index 000000000..5a50e094f --- /dev/null +++ b/previews/PR604/manual/representation/index.html @@ -0,0 +1,26 @@ + +Term representation and simplification · SymbolicUtils.jl
GitHub

Term representation and simplification

Performance of symbolic simplification depends on the datastructures used to represent terms. Efficient datastructures often have the advantage of automatic simplification, and of efficient storage.

The most basic term representation simply holds a function call and stores the function and the arguments it is called with. This is done by the Term type in SymbolicUtils. Functions that aren't commutative or associative, such as sin or hypot are stored as Terms. Commutative and associative operations like +, *, and their supporting operations like -, / and ^, when used on terms of type <:Number, stand to gain from the use of more efficient datastrucutres.

All term representations must support operation and arguments functions. And they must define istree to return true when called with an instance of the type. Generic term-manipulation programs such as the rule-based rewriter make use of this interface to inspect expressions. In this way, the interface wins back the generality lost by having a zoo of term representations instead of one. (see interface section for more on this.)

Preliminary representation of arithmetic

Linear combinations such as $\alpha_1 x_1 + \alpha_2 x_2 +...+ \alpha_n x_n$ are represented by Add(Dict(x₁ => α₁, x₂ => α₂, ..., xₙ => αₙ)). Here, any $x_i$ may itself be other types mentioned here, except for Add. When an Add is added to an Add, we merge their dictionaries and add up matching coefficients to create a single "flattened" Add.

Similarly, $x_1^{m_1}x_2^{m_2}...x_{m_n}$ is represented by Mul(Dict(x₁ => m₁, x₂ => m₂, ..., xₙ => mₙ)). $x_i$ may not themselves be Mul, multiplying a Mul with another Mul returns a "flattened" Mul.

Note that Add and Mul types perform a preliminary simplification which suffices to simplify numeric expressions to a large extent during construction.

\[p / q\]

is represented by Div(p, q). The result of * on Div is maintained as a Div. For example, Div(p_1, q_1) * Div(p_2, q_2) results in Div(p_1 * p_2, q_1 * q_2) and so on. The effect is, in Div(p, q), p or q or, if they are Mul, any of their multiplicands is not a Div. So Muls must always be nested inside a Div and can never show up immediately wrapping it. This rule sets up an expression so that it helps the simplify_fractions procedure described two sections below.

Polynomial representation

Packages like DynamicPolynomials.jl provide representations that are even more efficient than the Add and Mul types mentioned above. They are designed specifically for multi-variate polynomials. They provide common algorithms such as multi-variate polynomial GCD. The restrictions that make it fast also mean some things are not possible: Firstly, DynamicPolynomials can only represent flat polynomials. For example, (x-3)*(x+5) can only be represented as (x^2) + 15 - 8x. Secondly, DynamicPolynomials does not have ways to represent generic Terms such as sin(x-y) in the tree.

To reconcile these differences while being able to use the efficient algorithms of DynamicPolynomials we have the PolyForm type. This type holds a polynomial and the mappings necessary to present the polynomial as a SymbolicUtils expression (i.e. by defining operation and arguments). The mappings constructed for the conversion are 1) a bijection from DynamicPolynomials Variable type to a Symbolics Sym, and 2) a mapping from Syms to non-polynomial terms that the Syms stand-in for. These terms may themselves contain PolyForm if there are polynomials inside them. The mappings are transiently global, that is, when all references to the mappings go out of scope, they are released and re-created.

julia> @syms x y
+(x, y)
+
+julia> PolyForm((x-3)*(y-5))
+x*y + 15 - 5x - 3y

Terms for which the operation is not +, *, or ^ are replaced with a generated symbol when representing the polynomial, and a mapping from this new symbol to the original expression it stands-in for is maintained as stated above.

julia> p = PolyForm((sin(x) + cos(x))^2)
+(cos(x)^2) + 2cos(x)*sin(x) + (sin(x)^2)
+
+julia> p.p # this is the actual DynamicPolynomial stored
+cos_3658410937268741549² + 2cos_3658410937268741549sin_10720964503106793468 + sin_10720964503106793468²

By default, polynomials inside non-polynomial terms are not also converted to PolyForm. For example,

julia> PolyForm(sin((x-3)*(y-5)))
+sin((x - 3)*(y - 5))

But you can pass in the recurse=true keyword argument to do this.

julia> PolyForm(sin((x-3)*(y-5)), recurse=true)
+sin(x*y + 15 - 5x - 3y)

Polynomials are constructed by first turning symbols and non-polynomial terms into DynamicPolynomials-style variables and then applying the +, *, ^ operations on these variables. You can control the list of the polynomial operations with the Fs keyword argument. It is a Union type of the functions to apply. For example, let's say you want to turn terms into polynomials by only applying the + and ^ operations, and want to preserve * operations as-is, you could pass in Fs=Union{typeof(+), typeof(^)}

julia> PolyForm((x+y)^2*(x-y), Fs=Union{typeof(+), typeof(^)}, recurse=true)
+((x - (y))*((x^2) + 2x*y + (y^2)))

Note that in this case recurse=true was necessary as otherwise the polynomialization would not descend into the * operation as it is now considered a generic operation.

Simplifying fractions

simplify_fractions(expr) recurses through expr finding Divs and simplifying them using polynomial division.

First the factors of the numerators and the denominators are converted into PolyForm objects, then numerators and denominators are divided by their respective pairwise GCDs. The conversion of the numerator and denominator into PolyForm is set up so that simplify_fractions does not result in increase in the expression size due to polynomial expansion. Specifically, the factors are individually converted into PolyForm objects, and any powers of polynomial is not expanded, but the division process repeatedly divides them as many times as the power.

julia> simplify_fractions((x*y+5x+3y+15)/((x+3)*(x-4)))
+(5.0 + y) / (x - 4)
+
+julia> simplify_fractions((x*y+5x+3y+15)^2/((x+3)*(x-4)))
+((5.0 + y)*(15 + 5x + x*y + 3y)) / (x - 4)
+
+julia> simplify_fractions(3/(x+3) + x/(x+3))
+1
+
+julia> simplify_fractions(sin(x)/cos(x) + cos(x)/sin(x))
+(cos(x)^2 + sin(x)^2) / (cos(x)*sin(x))
+
+julia> simplify(ans)
+1 / (cos(x)*sin(x))
diff --git a/previews/PR604/manual/rewrite/index.html b/previews/PR604/manual/rewrite/index.html new file mode 100644 index 000000000..95408b588 --- /dev/null +++ b/previews/PR604/manual/rewrite/index.html @@ -0,0 +1,50 @@ + +Term Rewriting · SymbolicUtils.jl
GitHub

Term Rewriting

Rule-based rewriting

Rewrite rules match and transform an expression. A rule is written using either the @rule macro or the @acrule macro. It creates a callable Rule object.

Basics of rule-based term rewriting in SymbolicUtils

Here is a simple rewrite rule, that uses formula for the double angle of the sine function:

using SymbolicUtils
+
+@syms w z α::Real β::Real
+
+(w, z, α, β) # hide
+
+r1 = @rule sin(2(~x)) => 2sin(~x)*cos(~x)
+
+r1(sin(2z))

The @rule macro takes a pair of patterns – the matcher and the consequent (@rule matcher => consequent). If an expression matches the matcher pattern, it is rewritten to the consequent pattern. @rule returns a callable object that applies the rule to an expression.

~x in the example is what is a slot variable named x. In a matcher pattern, slot variables are placeholders that match exactly one expression. When used on the consequent side, they stand in for the matched expression. If a slot variable appears twice in a matcher pattern, all corresponding matches must be equal (as tested by Base.isequal function). Hence this rule says: if you see something added to itself, make it twice of that thing, and works as such.

If you try to apply this rule to an expression with triple angle, it will return nothing – this is the way a rule signifies failure to match.

r1(sin(3z)) === nothing

Slot variable (matcher) is not necessary a single variable

r1(sin(2*(w-z)))

but it must be a single expression

r1(sin(2*(w+z)*(α+β))) === nothing

Rules are of course not limited to single slot variable

r2 = @rule sin(~x + ~y) => sin(~x)*cos(~y) + cos(~x)*sin(~y);
+
+r2(sin(α+β))

If you want to match a variable number of subexpressions at once, you will need a segment variable. ~~xs in the following example is a segment variable:

@syms x y z
+@rule(+(~~xs) => ~~xs)(x + y + z)

~~xs is a vector of subexpressions matched. You can use it to construct something more useful:

r3 = @rule ~x * +(~~ys) => sum(map(y-> ~x * y, ~~ys));
+
+r3(2 * (w+w+α+β))

Notice that the expression was autosimplified before application of the rule.

2 * (w+w+α+β)

Predicates for matching

Matcher pattern may contain slot variables with attached predicates, written as ~x::f where f is a function that takes a matched expression and returns a boolean value. Such a slot will be considered a match only if f returns true.

Similarly ~~x::g is a way of attaching a predicate g to a segment variable. In the case of segment variables g gets a vector of 0 or more expressions and must return a boolean value. If the same slot or segment variable appears twice in the matcher pattern, then at most one of the occurrence should have a predicate.

For example,

@syms a b c d
+
+r = @rule ~x + ~~y::(ys->iseven(length(ys))) => "odd terms";
+
+@show r(a + b + c + d)
+@show r(b + c + d)
+@show r(b + c + b)
+@show r(a + b)

Associative-Commutative Rules

Given an expression f(x, f(y, z, u), v, w), a f is said to be associative if the expression is equivalent to f(x, y, z, u, v, w) and commutative if the order of arguments does not matter. SymbolicUtils has a special @acrule macro meant for rules on functions which are associate and commutative such as addition and multiplication of real and complex numbers.

@syms x y z
+
+acr = @acrule((~a)^(~x) * (~a)^(~y) => (~a)^(~x + ~y))
+
+acr(x^y * x^z)

although in case of Number it also works the same way with regular @rule since autosimplification orders and applies associativity and commutativity to the expression.

Example of applying the rules to simplify expression

Consider expression (cos(x) + sin(x))^2 that we would like simplify by applying some trigonometric rules. First, we need rule to expand square of cos(x) + sin(x). First we try the simplest rule to expand square of the sum and try it on simple expression

using SymbolicUtils
+
+@syms x::Real y::Real
+
+sqexpand = @rule (~x + ~y)^2 => (~x)^2 + (~y)^2 + 2 * ~x * ~y
+
+sqexpand((cos(x) + sin(x))^2)

It works. This can be further simplified using Pythagorean identity and check it

pyid = @rule sin(~x)^2 + cos(~x)^2 => 1
+
+pyid(cos(x)^2 + sin(x)^2) === nothing

Why does it return nothing? If we look at the rule, we see that the order of sin(x) and cos(x) is different. Therefore, in order to work, the rule needs to be associative-commutative.

acpyid = @acrule sin(~x)^2 + cos(~x)^2 => 1
+
+acpyid(cos(x)^2 + sin(x)^2 + 2cos(x)*sin(x))

It has been some work. Fortunately rules may be chained together into more sophisticated rewriters to avoid manual application of the rules.

Composing rewriters

A rewriter is any callable object which takes an expression and returns an expression or nothing. If nothing is returned that means there was no changes applicable to the input expression. The Rules we created above are rewriters.

The SymbolicUtils.Rewriters module contains some types which create and transform rewriters.

  • Empty() is a rewriter which always returns nothing
  • Chain(itr) chain an iterator of rewriters into a single rewriter which applies each chained rewriter in the given order. If a rewriter returns nothing this is treated as a no-change.
  • RestartedChain(itr) like Chain(itr) but restarts from the first rewriter once on the first successful application of one of the chained rewriters.
  • IfElse(cond, rw1, rw2) runs the cond function on the input, applies rw1 if cond returns true, rw2 if it returns false
  • If(cond, rw) is the same as IfElse(cond, rw, Empty())
  • Prewalk(rw; threaded=false, thread_cutoff=100) returns a rewriter which does a pre-order (from top to bottom and from left to right) traversal of a given expression and applies the rewriter rw. threaded=true will use multi threading for traversal. Note that if rw returns nothing when a match is not found, then Prewalk(rw) will also return nothing unless a match is found at every level of the walk. If you are applying multiple rules, then Chain already has the appropriate passthrough behavior. If you only want to apply one rule, then consider using PassThrough. thread_cutoff is the minimum number of nodes in a subtree which should be walked in a threaded spawn.
  • Postwalk(rw; threaded=false, thread_cutoff=100) similarly does post-order (from left to right and from bottom to top) traversal.
  • Fixpoint(rw) returns a rewriter which applies rw repeatedly until there are no changes to be made.
  • PassThrough(rw) returns a rewriter which if rw(x) returns nothing will instead return x otherwise will return rw(x).

Chaining rewriters

Several rules may be chained to give chain of rules. Chain is an array of rules which are subsequently applied to the expression.

To check that, we will combine rules from previous example into a chain

using SymbolicUtils
+using SymbolicUtils.Rewriters
+
+sqexpand = @rule (~x + ~y)^2 => (~x)^2 + (~y)^2 + 2 * ~x * ~y
+acpyid = @acrule sin(~x)^2 + cos(~x)^2 => 1
+
+csa = Chain([sqexpand, acpyid])
+
+csa((cos(x) + sin(x))^2)

Important feature of Chain is that it returns the expression instead of nothing if it doesn't change the expression

Chain([@acrule sin(~x)^2 + cos(~x)^2 => 1])((cos(x) + sin(x))^2)

it's important to notice, that chain is ordered, so if rules are in different order it wouldn't work the same as in earlier example

cas = Chain([acpyid, sqexpand])
+
+cas((cos(x) + sin(x))^2)

since Pythagorean identity is applied before square expansion, so it is unable to match squares of sine and cosine.

One way to circumvent the problem of order of applying rules in chain is to use RestartedChain

using SymbolicUtils.Rewriters: RestartedChain
+
+rcas = RestartedChain([acpyid, sqexpand])
+
+rcas((cos(x) + sin(x))^2)

It restarts the chain after each successful application of a rule, so after sqexpand is hit it (re)starts again and successfully applies acpyid to resulting expression.

You can also use Fixpoint to apply the rules until there are no changes.

Fixpoint(cas)((cos(x) + sin(x))^2)
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And they must define istree to return true when called with an instance of the type. Generic term-manipulation programs such as the rule-based rewriter make use of this interface to inspect expressions. In this way, the interface wins back the generality lost by having a zoo of term representations instead of one. (see interface section for more on this.)","category":"page"},{"location":"manual/representation/#Preliminary-representation-of-arithmetic","page":"Term representation and simplification","title":"Preliminary representation of arithmetic","text":"","category":"section"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"Linear combinations such as alpha_1 x_1 + alpha_2 x_2 ++ alpha_n x_n are represented by Add(Dict(x₁ => α₁, x₂ => α₂, ..., xₙ => αₙ)). Here, any x_i may itself be other types mentioned here, except for Add. When an Add is added to an Add, we merge their dictionaries and add up matching coefficients to create a single \"flattened\" Add.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"Similarly, x_1^m_1x_2^m_2x_m_n is represented by Mul(Dict(x₁ => m₁, x₂ => m₂, ..., xₙ => mₙ)). x_i may not themselves be Mul, multiplying a Mul with another Mul returns a \"flattened\" Mul.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"Note that Add and Mul types perform a preliminary simplification which suffices to simplify numeric expressions to a large extent during construction.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"p q","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"is represented by Div(p, q). The result of * on Div is maintained as a Div. For example, Div(p_1, q_1) * Div(p_2, q_2) results in Div(p_1 * p_2, q_1 * q_2) and so on. The effect is, in Div(p, q), p or q or, if they are Mul, any of their multiplicands is not a Div. So Muls must always be nested inside a Div and can never show up immediately wrapping it. This rule sets up an expression so that it helps the simplify_fractions procedure described two sections below.","category":"page"},{"location":"manual/representation/#Polynomial-representation","page":"Term representation and simplification","title":"Polynomial representation","text":"","category":"section"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"Packages like DynamicPolynomials.jl provide representations that are even more efficient than the Add and Mul types mentioned above. They are designed specifically for multi-variate polynomials. They provide common algorithms such as multi-variate polynomial GCD. The restrictions that make it fast also mean some things are not possible: Firstly, DynamicPolynomials can only represent flat polynomials. For example, (x-3)*(x+5) can only be represented as (x^2) + 15 - 8x. Secondly, DynamicPolynomials does not have ways to represent generic Terms such as sin(x-y) in the tree.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"To reconcile these differences while being able to use the efficient algorithms of DynamicPolynomials we have the PolyForm type. This type holds a polynomial and the mappings necessary to present the polynomial as a SymbolicUtils expression (i.e. by defining operation and arguments). The mappings constructed for the conversion are 1) a bijection from DynamicPolynomials Variable type to a Symbolics Sym, and 2) a mapping from Syms to non-polynomial terms that the Syms stand-in for. These terms may themselves contain PolyForm if there are polynomials inside them. The mappings are transiently global, that is, when all references to the mappings go out of scope, they are released and re-created.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"julia> @syms x y\n(x, y)\n\njulia> PolyForm((x-3)*(y-5))\nx*y + 15 - 5x - 3y","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"Terms for which the operation is not +, *, or ^ are replaced with a generated symbol when representing the polynomial, and a mapping from this new symbol to the original expression it stands-in for is maintained as stated above.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"julia> p = PolyForm((sin(x) + cos(x))^2)\n(cos(x)^2) + 2cos(x)*sin(x) + (sin(x)^2)\n\njulia> p.p # this is the actual DynamicPolynomial stored\ncos_3658410937268741549² + 2cos_3658410937268741549sin_10720964503106793468 + sin_10720964503106793468²","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"By default, polynomials inside non-polynomial terms are not also converted to PolyForm. For example,","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"julia> PolyForm(sin((x-3)*(y-5)))\nsin((x - 3)*(y - 5))","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"But you can pass in the recurse=true keyword argument to do this.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"julia> PolyForm(sin((x-3)*(y-5)), recurse=true)\nsin(x*y + 15 - 5x - 3y)","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"Polynomials are constructed by first turning symbols and non-polynomial terms into DynamicPolynomials-style variables and then applying the +, *, ^ operations on these variables. You can control the list of the polynomial operations with the Fs keyword argument. It is a Union type of the functions to apply. 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The conversion of the numerator and denominator into PolyForm is set up so that simplify_fractions does not result in increase in the expression size due to polynomial expansion. Specifically, the factors are individually converted into PolyForm objects, and any powers of polynomial is not expanded, but the division process repeatedly divides them as many times as the power.","category":"page"},{"location":"manual/representation/","page":"Term representation and simplification","title":"Term representation and simplification","text":"julia> simplify_fractions((x*y+5x+3y+15)/((x+3)*(x-4)))\n(5.0 + y) / (x - 4)\n\njulia> simplify_fractions((x*y+5x+3y+15)^2/((x+3)*(x-4)))\n((5.0 + y)*(15 + 5x + x*y + 3y)) / (x - 4)\n\njulia> simplify_fractions(3/(x+3) + x/(x+3))\n1\n\njulia> simplify_fractions(sin(x)/cos(x) + cos(x)/sin(x))\n(cos(x)^2 + sin(x)^2) / (cos(x)*sin(x))\n\njulia> simplify(ans)\n1 / (cos(x)*sin(x))","category":"page"},{"location":"manual/rewrite/#Term-Rewriting","page":"Term Rewriting","title":"Term Rewriting","text":"","category":"section"},{"location":"manual/rewrite/#Rule-based-rewriting","page":"Term Rewriting","title":"Rule-based rewriting","text":"","category":"section"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Rewrite rules match and transform an expression. A rule is written using either the @rule macro or the @acrule macro. It creates a callable Rule object.","category":"page"},{"location":"manual/rewrite/#Basics-of-rule-based-term-rewriting-in-SymbolicUtils","page":"Term Rewriting","title":"Basics of rule-based term rewriting in SymbolicUtils","text":"","category":"section"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Here is a simple rewrite rule, that uses formula for the double angle of the sine function:","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"using SymbolicUtils\n\n@syms w z α::Real β::Real\n\n(w, z, α, β) # hide\n\nr1 = @rule sin(2(~x)) => 2sin(~x)*cos(~x)\n\nr1(sin(2z))","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"The @rule macro takes a pair of patterns – the matcher and the consequent (@rule matcher => consequent). If an expression matches the matcher pattern, it is rewritten to the consequent pattern. @rule returns a callable object that applies the rule to an expression.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"~x in the example is what is a slot variable named x. In a matcher pattern, slot variables are placeholders that match exactly one expression. When used on the consequent side, they stand in for the matched expression. If a slot variable appears twice in a matcher pattern, all corresponding matches must be equal (as tested by Base.isequal function). Hence this rule says: if you see something added to itself, make it twice of that thing, and works as such.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"If you try to apply this rule to an expression with triple angle, it will return nothing – this is the way a rule signifies failure to match.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"r1(sin(3z)) === nothing","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Slot variable (matcher) is not necessary a single variable","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"r1(sin(2*(w-z)))","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"but it must be a single expression","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"r1(sin(2*(w+z)*(α+β))) === nothing","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Rules are of course not limited to single slot variable","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"r2 = @rule sin(~x + ~y) => sin(~x)*cos(~y) + cos(~x)*sin(~y);\n\nr2(sin(α+β))","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"If you want to match a variable number of subexpressions at once, you will need a segment variable. ~~xs in the following example is a segment variable:","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"@syms x y z\n@rule(+(~~xs) => ~~xs)(x + y + z)","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"~~xs is a vector of subexpressions matched. You can use it to construct something more useful:","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"r3 = @rule ~x * +(~~ys) => sum(map(y-> ~x * y, ~~ys));\n\nr3(2 * (w+w+α+β))","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Notice that the expression was autosimplified before application of the rule.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"2 * (w+w+α+β)","category":"page"},{"location":"manual/rewrite/#Predicates-for-matching","page":"Term Rewriting","title":"Predicates for matching","text":"","category":"section"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Matcher pattern may contain slot variables with attached predicates, written as ~x::f where f is a function that takes a matched expression and returns a boolean value. Such a slot will be considered a match only if f returns true.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Similarly ~~x::g is a way of attaching a predicate g to a segment variable. In the case of segment variables g gets a vector of 0 or more expressions and must return a boolean value. If the same slot or segment variable appears twice in the matcher pattern, then at most one of the occurrence should have a predicate.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"For example,","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"@syms a b c d\n\nr = @rule ~x + ~~y::(ys->iseven(length(ys))) => \"odd terms\";\n\n@show r(a + b + c + d)\n@show r(b + c + d)\n@show r(b + c + b)\n@show r(a + b)","category":"page"},{"location":"manual/rewrite/#Associative-Commutative-Rules","page":"Term Rewriting","title":"Associative-Commutative Rules","text":"","category":"section"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Given an expression f(x, f(y, z, u), v, w), a f is said to be associative if the expression is equivalent to f(x, y, z, u, v, w) and commutative if the order of arguments does not matter. SymbolicUtils has a special @acrule macro meant for rules on functions which are associate and commutative such as addition and multiplication of real and complex numbers.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"@syms x y z\n\nacr = @acrule((~a)^(~x) * (~a)^(~y) => (~a)^(~x + ~y))\n\nacr(x^y * x^z)","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"although in case of Number it also works the same way with regular @rule since autosimplification orders and applies associativity and commutativity to the expression.","category":"page"},{"location":"manual/rewrite/#Example-of-applying-the-rules-to-simplify-expression","page":"Term Rewriting","title":"Example of applying the rules to simplify expression","text":"","category":"section"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Consider expression (cos(x) + sin(x))^2 that we would like simplify by applying some trigonometric rules. First, we need rule to expand square of cos(x) + sin(x). First we try the simplest rule to expand square of the sum and try it on simple expression","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"using SymbolicUtils\n\n@syms x::Real y::Real\n\nsqexpand = @rule (~x + ~y)^2 => (~x)^2 + (~y)^2 + 2 * ~x * ~y\n\nsqexpand((cos(x) + sin(x))^2)","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"It works. This can be further simplified using Pythagorean identity and check it","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"pyid = @rule sin(~x)^2 + cos(~x)^2 => 1\n\npyid(cos(x)^2 + sin(x)^2) === nothing","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Why does it return nothing? If we look at the rule, we see that the order of sin(x) and cos(x) is different. Therefore, in order to work, the rule needs to be associative-commutative.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"acpyid = @acrule sin(~x)^2 + cos(~x)^2 => 1\n\nacpyid(cos(x)^2 + sin(x)^2 + 2cos(x)*sin(x))","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"It has been some work. Fortunately rules may be chained together into more sophisticated rewriters to avoid manual application of the rules.","category":"page"},{"location":"manual/rewrite/#Composing-rewriters","page":"Term Rewriting","title":"Composing rewriters","text":"","category":"section"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"A rewriter is any callable object which takes an expression and returns an expression or nothing. If nothing is returned that means there was no changes applicable to the input expression. The Rules we created above are rewriters.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"The SymbolicUtils.Rewriters module contains some types which create and transform rewriters.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Empty() is a rewriter which always returns nothing\nChain(itr) chain an iterator of rewriters into a single rewriter which applies each chained rewriter in the given order. If a rewriter returns nothing this is treated as a no-change.\nRestartedChain(itr) like Chain(itr) but restarts from the first rewriter once on the first successful application of one of the chained rewriters.\nIfElse(cond, rw1, rw2) runs the cond function on the input, applies rw1 if cond returns true, rw2 if it returns false\nIf(cond, rw) is the same as IfElse(cond, rw, Empty())\nPrewalk(rw; threaded=false, thread_cutoff=100) returns a rewriter which does a pre-order (from top to bottom and from left to right) traversal of a given expression and applies the rewriter rw. threaded=true will use multi threading for traversal. Note that if rw returns nothing when a match is not found, then Prewalk(rw) will also return nothing unless a match is found at every level of the walk. If you are applying multiple rules, then Chain already has the appropriate passthrough behavior. If you only want to apply one rule, then consider using PassThrough. thread_cutoff is the minimum number of nodes in a subtree which should be walked in a threaded spawn.\nPostwalk(rw; threaded=false, thread_cutoff=100) similarly does post-order (from left to right and from bottom to top) traversal.\nFixpoint(rw) returns a rewriter which applies rw repeatedly until there are no changes to be made.\nPassThrough(rw) returns a rewriter which if rw(x) returns nothing will instead return x otherwise will return rw(x).","category":"page"},{"location":"manual/rewrite/#Chaining-rewriters","page":"Term Rewriting","title":"Chaining rewriters","text":"","category":"section"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Several rules may be chained to give chain of rules. Chain is an array of rules which are subsequently applied to the expression.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"To check that, we will combine rules from previous example into a chain","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"using SymbolicUtils\nusing SymbolicUtils.Rewriters\n\nsqexpand = @rule (~x + ~y)^2 => (~x)^2 + (~y)^2 + 2 * ~x * ~y\nacpyid = @acrule sin(~x)^2 + cos(~x)^2 => 1\n\ncsa = Chain([sqexpand, acpyid])\n\ncsa((cos(x) + sin(x))^2)","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Important feature of Chain is that it returns the expression instead of nothing if it doesn't change the expression","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Chain([@acrule sin(~x)^2 + cos(~x)^2 => 1])((cos(x) + sin(x))^2)","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"it's important to notice, that chain is ordered, so if rules are in different order it wouldn't work the same as in earlier example","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"cas = Chain([acpyid, sqexpand])\n\ncas((cos(x) + sin(x))^2)","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"since Pythagorean identity is applied before square expansion, so it is unable to match squares of sine and cosine.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"One way to circumvent the problem of order of applying rules in chain is to use RestartedChain","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"using SymbolicUtils.Rewriters: RestartedChain\n\nrcas = RestartedChain([acpyid, sqexpand])\n\nrcas((cos(x) + sin(x))^2)","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"It restarts the chain after each successful application of a rule, so after sqexpand is hit it (re)starts again and successfully applies acpyid to resulting expression.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"You can also use Fixpoint to apply the rules until there are no changes.","category":"page"},{"location":"manual/rewrite/","page":"Term Rewriting","title":"Term Rewriting","text":"Fixpoint(cas)((cos(x) + sin(x))^2)","category":"page"},{"location":"upgrade/#Upgrade-to-SymbolicUtils-v1","page":"Upgrade to SymbolicUtils v1","title":"Upgrade to SymbolicUtils v1","text":"","category":"section"},{"location":"upgrade/","page":"Upgrade to SymbolicUtils v1","title":"Upgrade to SymbolicUtils v1","text":"The version 1 of SymbolicUtils utilizes Unityper to speed up the compilation time. We introduce a new type BasicSymbolic <: Symbolic to represent all the previous types Sym, Term, Add, Mul, Pow and Div. Since BasicSymbolic is a concrete type, checking x isa Sym or x isa Pow no longer works. Instead, one should use issym(x) or ispow(x) to check the \"type\" of the expression. Also, note that if one does not need to work on a specific symbolic representation, issym(x) and isterm(x) should be replaced by x isa Symbolic && !istree(x) and istree(x) to be more generic. Furthermore, dispatching on Sym, Term, etc no longer works. Instead, a function that takes BasicSymbolic should check the type of the expression using a if statement to select the right code path.","category":"page"},{"location":"upgrade/","page":"Upgrade to SymbolicUtils v1","title":"Upgrade to SymbolicUtils v1","text":"Although we don't have Sym, Term, etc in the Julia type system anymore, we still provide convenient constructors for them. For example, Sym{Real}(:x) would still work. For constructor that takes a collection like Terms, we recommend directly construct Any element type constructors like Term(f, Any[x1, x2]) to reduce extra memory allocation and compile time.","category":"page"},{"location":"manual/codegen/#Code-generation","page":"Code generation","title":"Code generation","text":"","category":"section"},{"location":"manual/codegen/","page":"Code generation","title":"Code generation","text":"Note: this feature is experimental and the API might change frequently","category":"page"},{"location":"manual/codegen/","page":"Code generation","title":"Code generation","text":"toexpr(ex) turns any expression (ex) into the equivalent Expr which is suitable for eval. The SymbolicUtils.Code module provides some combinators which provides the ability to construct more complex expressions than just function calls. These include:","category":"page"},{"location":"manual/codegen/","page":"Code generation","title":"Code generation","text":"Let blocks\nFunctions with arguments and keyword arguments\nFunctions with arguments which are to be de-structured\nExpressions that set array elements in-place\nExpressions that create an array similar in type to a reference array (currently supports Array, StaticArrays.SArray, and LabelledArrays.SLArray)\nExpressions that create sparse arrays","category":"page"},{"location":"manual/codegen/","page":"Code generation","title":"Code generation","text":"Do using SymbolicUtils.Code to get the following bindings","category":"page"},{"location":"manual/codegen/#toexpr","page":"Code generation","title":"toexpr","text":"","category":"section"},{"location":"manual/codegen/","page":"Code generation","title":"Code generation","text":"SymbolicUtils.Code.toexpr","category":"page"},{"location":"manual/codegen/#SymbolicUtils.Code.toexpr","page":"Code generation","title":"SymbolicUtils.Code.toexpr","text":"toexpr(ex, [st,])\n\nConvert a symbolic expression into an Expr, suitable to be passed into eval.\n\nFor example,\n\njulia> @syms a b\n(a, b)\n\njulia> toexpr(a+b)\n:((+)(a, b))\n\njulia> toexpr(a+b) |> dump\nExpr\n head: Symbol call\n args: Array{Any}((3,))\n 1: + (function of type typeof(+))\n 2: Symbol a\n 3: Symbol b\n\nNote that the function is an actual function object.\n\nFor more complex expressions, see other code-related combinators,\n\nNamely Assignment, Let, Func, SetArray, MakeArray, MakeSparseArray and MakeTuple.\n\nTo make your own type convertible to Expr using toexpr define toexpr(x, st) and forward the state st in internal calls to toexpr. st is state used to know when to leave something like y(t) as it is or when to make it var\"y(t)\". E.g. when y(t) is itself the argument of a function rather than y.\n\n\n\n\n\n","category":"function"},{"location":"manual/codegen/#Code-Combinators","page":"Code generation","title":"Code Combinators","text":"","category":"section"},{"location":"manual/codegen/","page":"Code generation","title":"Code generation","text":"These are all exported when you do using SymbolicUtils.Code","category":"page"},{"location":"manual/codegen/","page":"Code generation","title":"Code generation","text":"SymbolicUtils.Code.Assignment\nSymbolicUtils.Code.Let\nSymbolicUtils.Code.Func\nSymbolicUtils.Code.SpawnFetch\nSymbolicUtils.Code.SetArray\nSymbolicUtils.Code.MakeArray\nSymbolicUtils.Code.MakeSparseArray\nSymbolicUtils.Code.MakeTuple\nSymbolicUtils.Code.LiteralExpr","category":"page"},{"location":"manual/codegen/#SymbolicUtils.Code.Assignment","page":"Code generation","title":"SymbolicUtils.Code.Assignment","text":"Assignment(lhs, rhs)\n\nAn assignment expression. Shorthand lhs ← rhs (\\leftarrow)\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.Let","page":"Code generation","title":"SymbolicUtils.Code.Let","text":"Let(assignments, body[, let_block])\n\nA Let block.\n\nassignments is a vector of Assignments\nbody is the body of the let block\nlet_block boolean (default=true) – do not create a let block if false.\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.Func","page":"Code generation","title":"SymbolicUtils.Code.Func","text":"Func(args, kwargs, body[, pre])\n\nA function.\n\nargs is a vector of expressions\nkwargs is a vector of Assignments\nbody is the body of the function\npre a vector of expressions to be prepended to the function body, for example, it could be [Expr(:meta, :inline), Expr(:meta, :propagate_inbounds)] to create an @inline @propagate_inbounds function definition.\n\nSpecial features in args:\n\nargs can contain DestructuredArgs\ncall expressions\n\nFor example,\n\n\njulia> @syms a b c t f(d) x(t) y(t) z(t)\n(a, b, c, t, f(::Number)::Number, x(::Number)::Number, y(::Number)::Number, z(::Number)::Number)\n\njulia> func = Func([a,x(t), DestructuredArgs([b, y(t)]), f], # args\n [c ← 2, z(t) ← 42], # kwargs\n f((a + b + c) / x(t) + y(t) + z(t)));\n\njulia> toexpr(func)\n:(function (a, var\"x(t)\", var\"##arg#255\", f; c = 2, var\"z(t)\" = 42)\n let b = var\"##arg#255\"[1], var\"y(t)\" = var\"##arg#255\"[2]\n f((+)(var\"y(t)\", var\"z(t)\", (*)((+)(a, b, c), (inv)(var\"x(t)\"))))\n end\n end)\n\nthe second argument is a DestructuredArgs, in the Expr form, it is given a random name, and is expected to receive a vector or tuple of size 2 containing the values of b and y(t). The let block that is automatically generated \"destructures\" these arguments.\nx(t) and y(t) have been replaced with var\"x(t)\" and var\"y(t)\" symbols throughout\n\nthe generated Expr. This makes sure that we are not actually calling the expressions x(t) or y(t) but instead passing the right values in place of the whole expression.\n\nf is also a function-like symbol, same as x and y, but since the args array contains f as itself rather than as say, f(t), it does not become a var\"f(t)\". The generated function expects a function of one argument to be passed in the position of f.\n\nAn example invocation of this function is:\n\njulia> executable = eval(toexpr(func))\n#10 (generic function with 1 method)\n\njulia> executable(1, 2.0, [2,3.0], x->string(x); var\"z(t)\" = sqrt(42))\n\"11.98074069840786\"\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.SpawnFetch","page":"Code generation","title":"SymbolicUtils.Code.SpawnFetch","text":"SpawnFetch{ParallelType}(funcs [, args], reduce)\n\nRun every expression in funcs in its own task, the expression should be a Func object and is passed to Threads.Task(f). If Func takes arguments, then the arguments must be passed in as args–a vector of vector of arguments to each function in funcs. We don't use @spawn in order to support RuntimeGeneratedFunctions which disallow closures, instead we interpolate these functions or closures as smaller RuntimeGeneratedFunctions.\n\nreduce function is used to combine the results of executing exprs. A SpawnFetch expression returns the reduced result.\n\nUse Symbolics.MultithreadedForm ParallelType from the Symbolics.jl package to get the RuntimeGeneratedFunction version SpawnFetch.\n\nParallelType can be used to define more parallelism types SymbolicUtils supports Multithreaded type. Which spawns threaded tasks.\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.SetArray","page":"Code generation","title":"SymbolicUtils.Code.SetArray","text":"SetArray(inbounds, arr, elems)\n\nAn expression representing setting of elements of arr.\n\nBy default, every element of elems is copied over to arr,\n\nbut if elems contains AtIndex(i, val) objects, then arr[i] = val is performed in its place.\n\ninbounds is a boolean flag, true surrounds the resulting expression in an @inbounds.\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.MakeArray","page":"Code generation","title":"SymbolicUtils.Code.MakeArray","text":"MakeArray(elems, similarto, [output_eltype=nothing])\n\nAn expression which constructs an array.\n\nelems is the output array\nsimilarto can either be a type, or some symbol that is an array whose type needs to be emulated. If similarto is a StaticArrays.SArray, then the output array is also created as an SArray, similarly, an Array will result in an Array, and a LabelledArrays.SLArray will result in a labelled static array.\noutput_eltype: if set, then forces the element type of the output array to be this. by default, the output type is inferred automatically.\n\nYou can define:\n\n@inline function create_array(A::Type{<:MyArray},a\n ::Nothing, d::Val{dims}, elems...) where dims\n\n# and\n\n@inline function create_array(::Type{<:MyArray}, T, ::Val{dims}, elems...) where dims\n\nwhich creates an array of size dims using the elements elems and eltype T, to allow MakeArray to create arrays similarto MyArrays.\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.MakeSparseArray","page":"Code generation","title":"SymbolicUtils.Code.MakeSparseArray","text":"MakeSpaseArray(array)\n\nAn expression which creates a SparseMatrixCSC or a SparseVector.\n\nThe generated expression contains the sparsity information of array,\n\nit only creates the nzval field at run time.\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.MakeTuple","page":"Code generation","title":"SymbolicUtils.Code.MakeTuple","text":"MakeTuple(tup)\n\nMake a Tuple from a tuple of expressions.\n\n\n\n\n\n","category":"type"},{"location":"manual/codegen/#SymbolicUtils.Code.LiteralExpr","page":"Code generation","title":"SymbolicUtils.Code.LiteralExpr","text":"LiteralExpr(ex)\n\nLiterally ex, an Expr. toexpr on LiteralExpr recursively calls toexpr on any interpolated symbolic expressions.\n\n\n\n\n\n","category":"type"},{"location":"#SymbolicUtils.jl-—-Symbolic-programming-in-Julia","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"","category":"section"},{"location":"#Features","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Features","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Fast expressions\nA rule-based rewriting language.\nA combinator library for making rewriters.\nEfficient representation of numeric expressions\nType promotion:\nSymbols (Syms) carry type information. (read more)\nCompound expressions composed of Syms propagate type information. (read more)\nSet of extendable simplification rules.","category":"page"},{"location":"#Creating-symbolic-expressions","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Creating symbolic expressions","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"First, let's use the @syms macro to create a few symbols.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"using SymbolicUtils\n\n@syms w z α::Real β::Real\n\n(w, z, α, β) # hide\n","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Type annotations are optional when creating symbols. Here α, β behave like Real numbers. w and z behave like Number, which is the default. You can use the symtype function to find the type of a symbol.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"using SymbolicUtils: symtype\n\nsymtype(w), symtype(z), symtype(α), symtype(β)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Note however that they are not subtypes of these types!","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"@show w isa Number\n@show α isa Real","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"As their types are different:","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"@show typeof(w)\n@show typeof(α)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"(see this post for why they are all not just subtypes of Number)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"You can do basic arithmetic on symbols to get symbolic expressions:","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"expr1 = α*sin(w)^2 + β*cos(z)^2\nexpr2 = α*cos(z)^2 + β*sin(w)^2\n\nexpr1 + expr2","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"SymbolicUtils automatically simplifies","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"2w + 3w - 3z + α","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"and reorders","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"(z + w)*(α + β)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"expressions of type Symbolic{<:Number} (which includes Sym{Real}) when they are created. It also does constant elimination (including rational numbers)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"5 + 2w - 3z + α - (β + 5//3) + 3w - 2 + 3//2 * β","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"It's worth remembering that the expression may be transformed with respect to the input when it's created.","category":"page"},{"location":"#Function-like-symbols","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Function-like symbols","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Symbols can be defined to behave like functions. Both the input and output types for the function can be specified. Any application to that function will only admit either values of those types or symbols of the same symtype.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"using SymbolicUtils\n@syms f(x) g(x::Real, y::Real)::Real\n\nf(z) + g(1, α) + sin(w)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"This does not work since z is a Number, not a Real.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"g(1, z)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"This works because g \"returns\" a Real.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"g(2//5, g(1, β))","category":"page"},{"location":"#Expression-interface","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Expression interface","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Symbolic expressions are of type Term{T}, Add{T}, Mul{T}, Pow{T} or Div{T} and denote some function call where one or more arguments are themselves such expressions or Syms. See more about the representation here.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"All the expression types support the TermInterface.jl interface. Please refer to the package for the complete reference of the interface.","category":"page"},{"location":"#Term-rewriting","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Term rewriting","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"SymbolicUtils contains a rule-based rewriting language for easy pattern matching and rewriting of expression. There is also a combinator library to combine rules to chain, branch and loop over rules.","category":"page"},{"location":"#Simplification","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Simplification","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"By default +, * and ^ operations apply the most basic simplification upon construction of the expression.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"The rules with which the canonical form of Symbolic{<:Number} terms are constructed are the next (where x isa Symbolic and c isa Number)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"0 + x, 1 * x and x^1 always gives x\n0 * x always gives 0 and x ^ 0 gives 1\n-x becomes (-1)*x\ncommutativity and associativity over + and * are assumed. Re-ordering of terms will be done under a total order\np/q * x or x * p/q results in (p*x)/q\np/q * x/y results in (p*x)/(q*y)\nx + ... + x will be fused into n*x with type Mul\nx * ... * x will be fused into x^n with type Pow\nsum of Add's are fused\nproduct of Mul's are fused\nc * (c₁x₁ + ... + cₙxₙ) will be converted into c*c₁*x₁ + ... + c*cₙ*xₙ\n(x₁^c₁ * ... * xₙ^cₙ)^c will be converted into x₁^(c*c₁) * ... * xₙ^(c*cₙ)\nthere are come other simplifications on construction that you can check here","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Here is an example of this","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"2 * (w+w+α+β + sin(z)^2 + cos(z)^2 - 1)","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"The simplify function applies a built-in set of rules to rewrite expressions in order to simplify it.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"simplify(2 * (w+w+α+β + sin(z)^2 + cos(z)^2 - 1))","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"The rules in the default simplify applies simple constant elimination and trigonometric identities.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"If you read the previous section on the rules DSL, you should be able to read and understand the rules that are used by simplify.","category":"page"},{"location":"#Code-generation","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Code generation","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Experimental feature","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"It is common to want to generate executable code from symbolic expressions and blocks of them. We are working on experimental support for turning Symbolic expressions into executable functions with specific focus on array input and output and performance which is critical to the Differential Equations ecosystem which is making heavy use of this package.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"See Code generation for more about this.","category":"page"},{"location":"#Learn-more","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"Learn more","text":"","category":"section"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"If you have a package that you would like to utilize rule-based rewriting in, look at the suggestions in the Interfacing section to find out how you can do that without any fundamental changes to your package. Look at the API documentation for docstrings about specific functions or macros.","category":"page"},{"location":"","page":"SymbolicUtils.jl — Symbolic programming in Julia","title":"SymbolicUtils.jl — Symbolic programming in Julia","text":"Head over to the github repository to ask questions and report problems! Join the Zulip stream to chat!","category":"page"},{"location":"manual/interface/#Interfacing-with-SymbolicUtils.jl","page":"Interfacing with SymbolicUtils.jl","title":"Interfacing with SymbolicUtils.jl","text":"","category":"section"},{"location":"manual/interface/","page":"Interfacing with SymbolicUtils.jl","title":"Interfacing with SymbolicUtils.jl","text":"This section is for Julia package developers who may want to use the simplify and rule rewriting system on their own expression types.","category":"page"},{"location":"manual/interface/#Defining-the-interface","page":"Interfacing with SymbolicUtils.jl","title":"Defining the interface","text":"","category":"section"},{"location":"manual/interface/","page":"Interfacing with SymbolicUtils.jl","title":"Interfacing with SymbolicUtils.jl","text":"SymbolicUtils matchers can match any Julia object that implements an interface to traverse it as a tree. The interface in question, is defined in the TermInterface.jl package. Its purpose is to provide a shared interface between various symbolic programming Julia packages. ","category":"page"},{"location":"manual/interface/","page":"Interfacing with SymbolicUtils.jl","title":"Interfacing with SymbolicUtils.jl","text":"In particular, you should define methods from TermInterface.jl for an expression tree type T with symbol types S to work with SymbolicUtils.jl","category":"page"},{"location":"manual/interface/","page":"Interfacing with SymbolicUtils.jl","title":"Interfacing with SymbolicUtils.jl","text":"You can read the documentation of TermInterface.jl on the Github repository.","category":"page"},{"location":"manual/interface/#SymbolicUtils.jl-only-methods","page":"Interfacing with SymbolicUtils.jl","title":"SymbolicUtils.jl only methods","text":"","category":"section"},{"location":"manual/interface/","page":"Interfacing with SymbolicUtils.jl","title":"Interfacing with SymbolicUtils.jl","text":"promote_symtype(f, arg_symtypes...)","category":"page"},{"location":"manual/interface/","page":"Interfacing with SymbolicUtils.jl","title":"Interfacing with SymbolicUtils.jl","text":"Returns the appropriate output type of applying f on arguments of type arg_symtypes.","category":"page"}] +} diff --git a/previews/PR604/siteinfo.js b/previews/PR604/siteinfo.js new file mode 100644 index 000000000..3684ef64a --- /dev/null +++ b/previews/PR604/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR604"; diff --git a/previews/PR604/upgrade/index.html b/previews/PR604/upgrade/index.html new file mode 100644 index 000000000..e0a8b3474 --- /dev/null +++ b/previews/PR604/upgrade/index.html @@ -0,0 +1,2 @@ + +Upgrade to SymbolicUtils v1 · SymbolicUtils.jl
GitHub

Upgrade to SymbolicUtils v1

The version 1 of SymbolicUtils utilizes Unityper to speed up the compilation time. We introduce a new type BasicSymbolic <: Symbolic to represent all the previous types Sym, Term, Add, Mul, Pow and Div. Since BasicSymbolic is a concrete type, checking x isa Sym or x isa Pow no longer works. Instead, one should use issym(x) or ispow(x) to check the "type" of the expression. Also, note that if one does not need to work on a specific symbolic representation, issym(x) and isterm(x) should be replaced by x isa Symbolic && !istree(x) and istree(x) to be more generic. Furthermore, dispatching on Sym, Term, etc no longer works. Instead, a function that takes BasicSymbolic should check the type of the expression using a if statement to select the right code path.

Although we don't have Sym, Term, etc in the Julia type system anymore, we still provide convenient constructors for them. For example, Sym{Real}(:x) would still work. For constructor that takes a collection like Terms, we recommend directly construct Any element type constructors like Term(f, Any[x1, x2]) to reduce extra memory allocation and compile time.