mirror of
https://github.com/josdejong/mathjs.git
synced 2026-01-18 14:59:29 +00:00
672 lines
22 KiB
JavaScript
672 lines
22 KiB
JavaScript
'use strict';
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function factory (type, config, load, typed, math) {
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var parse = load(require('../../expression/parse'));
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var ConstantNode = load(require('../../expression/node/ConstantNode'));
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var FunctionNode = load(require('../../expression/node/FunctionNode'));
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var OperatorNode = load(require('../../expression/node/OperatorNode'));
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var ParenthesisNode = load(require('../../expression/node/ParenthesisNode'));
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var SymbolNode = load(require('../../expression/node/SymbolNode'));
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var Node = load(require('../../expression/node/Node'));
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var simplifyConstant = load(require('./simplify/simplifyConstant'));
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var simplifyCore = load(require('./simplify/simplifyCore'));
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var resolve = load(require('./simplify/resolve'));
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var util = load(require('./simplify/util'));
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var isCommutative = util.isCommutative;
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var isAssociative = util.isAssociative;
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var flatten = util.flatten;
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var unflattenr = util.unflattenr;
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var unflattenl = util.unflattenl;
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var createMakeNodeFunction = util.createMakeNodeFunction;
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/**
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* Simplify an expression tree.
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*
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* A list of rules are applied to an expression, repeating over the list until
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* no further changes are made.
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* It's possible to pass a custom set of rules to the function as second
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* argument. A rule can be specified as an object, string, or function:
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*
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* var rules = [
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* { l: 'n1*n3 + n2*n3', r: '(n1+n2)*n3' },
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* 'n1*n3 + n2*n3 -> (n1+n2)*n3',
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* function (node) {
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* // ... return a new node or return the node unchanged
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* return node
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* }
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* ]
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*
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* String and object rules consist of a left and right pattern. The left is
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* used to match against the expression and the right determines what matches
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* are replaced with. The main difference between a pattern and a normal
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* expression is that variables starting with the following characters are
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* interpreted as wildcards:
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*
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* - 'n' - matches any Node
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* - 'c' - matches any ConstantNode
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* - 'v' - matches any Node that is not a ConstantNode
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*
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* The default list of rules is exposed on the function as `simplify.rules`
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* and can be used as a basis to built a set of custom rules.
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*
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* For more details on the theory, see:
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*
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* - [Strategies for simplifying math expressions (Stackoverflow)](http://stackoverflow.com/questions/7540227/strategies-for-simplifying-math-expressions)
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* - [Symbolic computation - Simplification (Wikipedia)](https://en.wikipedia.org/wiki/Symbolic_computation#Simplification)
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*
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* Syntax:
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*
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* simplify(expr)
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* simplify(expr, rules)
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* simplify(expr, rules, scope)
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* simplify(expr, scope)
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*
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* Examples:
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*
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* math.simplify('2 * 1 * x ^ (2 - 1)'); // Node {2 * x}
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* math.simplify('2 * 3 * x', {x: 4}); // Node {24}
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* var f = math.parse('2 * 1 * x ^ (2 - 1)');
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* math.simplify(f); // Node {2 * x}
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*
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* See also:
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*
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* derivative, parse, eval
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*
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* @param {Node | string} expr
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* The expression to be simplified
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* @param {Array<{l:string, r: string} | string | function>} [rules]
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* Optional list with custom rules
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* @return {Node} Returns the simplified form of `expr`
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*/
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var simplify = typed('simplify', {
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'string': function (expr) {
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return simplify(parse(expr), simplify.rules, {});
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},
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'string, Object': function (expr, scope) {
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return simplify(parse(expr), simplify.rules, scope);
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},
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'string, Array': function (expr, rules) {
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return simplify(parse(expr), rules, {});
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},
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'string, Array, Object': function (expr, rules, scope) {
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return simplify(parse(expr), rules, scope);
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},
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'Node, Object': function (expr, scope) {
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return simplify(expr, simplify.rules, scope);
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},
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'Node': function (expr) {
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return simplify(expr, simplify.rules, {});
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},
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'Node, Array': function (expr, rules) {
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return simplify(expr, rules, {});
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},
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'Node, Array, Object': function (expr, rules, scope) {
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rules = _buildRules(rules);
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var res = resolve(expr, scope);
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var res = removeParens(res);
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var visited = {};
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var str = res.toString({parenthesis: 'all'});
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while(!visited[str]) {
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visited[str] = true;
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_lastsym = 0; // counter for placeholder symbols
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for (var i=0; i<rules.length; i++) {
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if (typeof rules[i] === 'function') {
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res = rules[i](res);
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}
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else {
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flatten(res);
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res = applyRule(res, rules[i]);
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}
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unflattenl(res); // using left-heavy binary tree here since custom rule functions may expect it
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}
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str = res.toString({parenthesis: 'all'});
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}
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return res;
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}
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});
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simplify.simplifyCore = simplifyCore;
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simplify.resolve = resolve;
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function removeParens(node) {
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return node.transform(function(node, path, parent) {
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return type.isParenthesisNode(node)
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? node.content
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: node;
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});
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}
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// All constants that are allowed in rules
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var SUPPORTED_CONSTANTS = {
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true: true,
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false: true,
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e: true,
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i: true,
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Infinity: true,
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LN2: true,
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LN10: true,
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LOG2E: true,
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LOG10E: true,
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NaN: true,
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phi: true,
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pi: true,
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SQRT1_2: true,
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SQRT2: true,
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tau: true,
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// null: false,
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// uninitialized: false,
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// version: false,
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};
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// Array of strings, used to build the ruleSet.
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// Each l (left side) and r (right side) are parsed by
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// the expression parser into a node tree.
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// Left hand sides are matched to subtrees within the
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// expression to be parsed and replaced with the right
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// hand side.
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// TODO: Add support for constraints on constants (either in the form of a '=' expression or a callback [callback allows things like comparing symbols alphabetically])
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// To evaluate lhs constants for rhs constants, use: { l: 'c1+c2', r: 'c3', evaluate: 'c3 = c1 + c2' }. Multiple assignments are separated by ';' in block format.
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// It is possible to get into an infinite loop with conflicting rules
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simplify.rules = [
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simplifyCore,
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//{ l: 'n+0', r: 'n' }, // simplifyCore
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//{ l: 'n^0', r: '1' }, // simplifyCore
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//{ l: '0*n', r: '0' }, // simplifyCore
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//{ l: 'n/n', r: '1'}, // simplifyCore
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//{ l: 'n^1', r: 'n' }, // simplifyCore
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//{ l: '+n1', r:'n1' }, // simplifyCore
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//{ l: 'n--n1', r:'n+n1' }, // simplifyCore
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{ l: 'log(e)', r:'1' },
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// temporary rules
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{ l: 'n-n1', r:'n+-n1' }, // temporarily replace 'subtract' so we can further flatten the 'add' operator
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{ l: '-(c*v)', r: '(-c) * v' }, // make non-constant terms positive
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{ l: '-v', r: '(-1) * v' },
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{ l: 'n/n1^n2', r:'n*n1^-n2' }, // temporarily replace 'divide' so we can further flatten the 'multiply' operator
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{ l: 'n/n1', r:'n*n1^-1' },
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// expand nested exponentiation
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{ l: '(n ^ n1) ^ n2', r: 'n ^ (n1 * n2)'},
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// collect like factors
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{ l: 'n*n', r: 'n^2' },
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{ l: 'n * n^n1', r: 'n^(n1+1)' },
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{ l: 'n^n1 * n^n2', r: 'n^(n1+n2)' },
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// collect like terms
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{ l: 'n+n', r: '2*n' },
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{ l: 'n+-n', r: '0' },
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{ l: 'n1*n2 + n2', r: '(n1+1)*n2' },
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{ l: 'n1*n3 + n2*n3', r: '(n1+n2)*n3' },
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// remove parenthesis in the case of negating a quantitiy
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{ l: 'n1 + -1 * (n2 + n3)', r:'n1 + -1 * n2 + -1 * n3' },
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simplifyConstant,
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{ l: '(-n)*n1', r: '-(n*n1)' }, // make factors positive (and undo 'make non-constant terms positive')
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// ordering of constants
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{ l: 'c+v', r: 'v+c', context: { 'add': { commutative:false } } },
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{ l: 'v*c', r: 'c*v', context: { 'multiply': { commutative:false } } },
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// undo temporary rules
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//{ l: '(-1) * n', r: '-n' }, // #811 added test which proved this is redundant
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{ l: 'n+-n1', r:'n-n1' }, // undo replace 'subtract'
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{ l: 'n*(n1^-1)', r:'n/n1' }, // undo replace 'divide'
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{ l: 'n*n1^-n2', r:'n/n1^n2' },
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{ l: 'n1^-1', r:'1/n1' },
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{ l: 'n*(n1/n2)', r:'(n*n1)/n2' }, // '*' before '/'
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{ l: 'n-(n1+n2)', r:'n-n1-n2' }, // '-' before '+'
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// { l: '(n1/n2)/n3', r: 'n1/(n2*n3)' },
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// { l: '(n*n1)/(n*n2)', r: 'n1/n2' },
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{ l: '1*n', r: 'n' } // this pattern can be produced by simplifyConstant
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];
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/**
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* Parse the string array of rules into nodes
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*
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* Example syntax for rules:
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*
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* Position constants to the left in a product:
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* { l: 'n1 * c1', r: 'c1 * n1' }
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* n1 is any Node, and c1 is a ConstantNode.
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*
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* Apply difference of squares formula:
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* { l: '(n1 - n2) * (n1 + n2)', r: 'n1^2 - n2^2' }
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* n1, n2 mean any Node.
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*
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* Short hand notation:
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* 'n1 * c1 -> c1 * n1'
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*/
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function _buildRules(rules) {
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// Array of rules to be used to simplify expressions
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var ruleSet = [];
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for(var i=0; i<rules.length; i++) {
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var rule = rules[i];
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var newRule;
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var ruleType = typeof rule;
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switch (ruleType) {
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case 'string':
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var lr = rule.split('->');
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if (lr.length !== 2) {
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throw SyntaxError('Could not parse rule: ' + rule);
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}
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rule = {l: lr[0], r: lr[1]};
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/* falls through */
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case 'object':
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newRule = {
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l: removeParens(parse(rule.l)),
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r: removeParens(parse(rule.r)),
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}
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if(rule.context) {
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newRule.evaluate = rule.context;
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}
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if(rule.evaluate) {
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newRule.evaluate = parse(rule.evaluate);
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}
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if (isAssociative(newRule.l)) {
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var makeNode = createMakeNodeFunction(newRule.l);
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var expandsym = _getExpandPlaceholderSymbol();
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newRule.expanded = {};
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newRule.expanded.l = makeNode([newRule.l.clone(), expandsym]);
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// Push the expandsym into the deepest possible branch.
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// This helps to match the newRule against nodes returned from getSplits() later on.
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flatten(newRule.expanded.l);
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unflattenr(newRule.expanded.l);
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newRule.expanded.r = makeNode([newRule.r, expandsym]);
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}
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break;
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case 'function':
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newRule = rule;
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break;
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default:
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throw TypeError('Unsupported type of rule: ' + ruleType);
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}
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// console.log('Adding rule: ' + rules[i]);
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// console.log(newRule);
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ruleSet.push(newRule);
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}
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return ruleSet;
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}
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var _lastsym = 0;
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function _getExpandPlaceholderSymbol() {
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return new SymbolNode('_p' + _lastsym++);
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}
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/**
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* Returns a simplfied form of node, or the original node if no simplification was possible.
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*
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* @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
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* @return {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} The simplified form of `expr`, or the original node if no simplification was possible.
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*/
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var applyRule = typed('applyRule', {
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'Node, Object': function (node, rule) {
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//console.log('Entering applyRule(' + node.toString() + ')');
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// Do not clone node unless we find a match
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var res = node;
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// First replace our child nodes with their simplified versions
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// If a child could not be simplified, the assignments will have
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// no effect since the node is returned unchanged
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if (res instanceof OperatorNode || res instanceof FunctionNode) {
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if (res.args) {
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for(var i=0; i<res.args.length; i++) {
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res.args[i] = applyRule(res.args[i], rule);
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}
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}
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}
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else if(res instanceof ParenthesisNode) {
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if(res.content) {
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res.content = applyRule(res.content, rule);
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}
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}
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// Try to match a rule against this node
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var repl = rule.r;
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var matches = _ruleMatch(rule.l, res)[0];
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// If the rule is associative operator, we can try matching it while allowing additional terms.
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// This allows us to match rules like 'n+n' to the expression '(1+x)+x' or even 'x+1+x' if the operator is commutative.
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if (!matches && rule.expanded) {
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repl = rule.expanded.r;
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matches = _ruleMatch(rule.expanded.l, res)[0];
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}
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if (matches) {
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// var before = res.toString({parenthesis: 'all'});
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// Create a new node by cloning the rhs of the matched rule
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res = repl.clone();
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// Replace placeholders with their respective nodes without traversing deeper into the replaced nodes
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var _transform = function(node) {
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if(node.isSymbolNode && matches.placeholders.hasOwnProperty(node.name)) {
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return matches.placeholders[node.name].clone();
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}
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else {
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return node.map(_transform);
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}
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}
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res = _transform(res);
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// var after = res.toString({parenthesis: 'all'});
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// console.log('Simplified ' + before + ' to ' + after);
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}
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return res;
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}
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});
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/**
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* Get (binary) combinations of a flattened binary node
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* e.g. +(node1, node2, node3) -> [
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* +(node1, +(node2, node3)),
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* +(node2, +(node1, node3)),
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* +(node3, +(node1, node2))]
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*
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*/
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function getSplits(node, context) {
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var res = [];
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var right, rightArgs;
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var makeNode = createMakeNodeFunction(node);
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if (isCommutative(node, context)) {
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for (var i=0; i<node.args.length; i++) {
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rightArgs = node.args.slice(0);
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rightArgs.splice(i, 1);
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right = (rightArgs.length === 1) ? rightArgs[0] : makeNode(rightArgs);
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res.push(makeNode([node.args[i], right]));
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}
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}
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else {
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rightArgs = node.args.slice(1);
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right = (rightArgs.length === 1) ? rightArgs[0] : makeNode(rightArgs);
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res.push(makeNode([node.args[0], right]));
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}
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return res;
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}
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/**
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* Returns the set union of two match-placeholders or null if there is a conflict.
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*/
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function mergeMatch(match1, match2) {
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var res = {placeholders:{}};
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// Some matches may not have placeholders; this is OK
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if (!match1.placeholders && !match2.placeholders) {
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return res;
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}
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else if (!match1.placeholders) {
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return match2;
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}
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else if (!match2.placeholders) {
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return match1;
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}
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// Placeholders with the same key must match exactly
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for (var key in match1.placeholders) {
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res.placeholders[key] = match1.placeholders[key];
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if (match2.placeholders.hasOwnProperty(key)) {
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if (!_exactMatch(match1.placeholders[key], match2.placeholders[key] )) {
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return null;
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}
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}
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}
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for (var key in match2.placeholders) {
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res.placeholders[key] = match2.placeholders[key];
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}
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return res;
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}
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/**
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* Combine two lists of matches by applying mergeMatch to the cartesian product of two lists of matches.
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* Each list represents matches found in one child of a node.
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*/
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function combineChildMatches(list1, list2) {
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var res = [];
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if (list1.length === 0 || list2.length === 0) {
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return res;
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}
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var merged;
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for (var i1 = 0; i1 < list1.length; i1++) {
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for (var i2 = 0; i2 < list2.length; i2++) {
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merged = mergeMatch(list1[i1], list2[i2]);
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if (merged) {
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res.push(merged);
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}
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}
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}
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return res;
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}
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/**
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* Combine multiple lists of matches by applying mergeMatch to the cartesian product of two lists of matches.
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* Each list represents matches found in one child of a node.
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* Returns a list of unique matches.
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*/
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function mergeChildMatches(childMatches) {
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if (childMatches.length === 0) {
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return childMatches;
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}
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var sets = childMatches.reduce(combineChildMatches);
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var uniqueSets = [];
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var unique = {};
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for(var i = 0; i < sets.length; i++) {
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var s = JSON.stringify(sets[i]);
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if (!unique[s]) {
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unique[s] = true;
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uniqueSets.push(sets[i]);
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}
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}
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return uniqueSets;
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}
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/**
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* Determines whether node matches rule.
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*
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* @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} rule
|
|
* @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
|
|
* @return {Object} Information about the match, if it exists.
|
|
*/
|
|
function _ruleMatch(rule, node, isSplit) {
|
|
// console.log('Entering _ruleMatch(' + JSON.stringify(rule) + ', ' + JSON.stringify(node) + ')');
|
|
// console.log('rule = ' + rule);
|
|
// console.log('node = ' + node);
|
|
|
|
// console.log('Entering _ruleMatch(' + rule.toString() + ', ' + node.toString() + ')');
|
|
var res = [{placeholders:{}}];
|
|
|
|
if (rule instanceof OperatorNode && node instanceof OperatorNode
|
|
|| rule instanceof FunctionNode && node instanceof FunctionNode) {
|
|
|
|
// If the rule is an OperatorNode or a FunctionNode, then node must match exactly
|
|
if (rule instanceof OperatorNode) {
|
|
if (rule.op !== node.op || rule.fn !== node.fn) {
|
|
return [];
|
|
}
|
|
}
|
|
else if (rule instanceof FunctionNode) {
|
|
if (rule.name !== node.name) {
|
|
return [];
|
|
}
|
|
}
|
|
|
|
// rule and node match. Search the children of rule and node.
|
|
if (node.args.length === 1 && rule.args.length === 1 || !isAssociative(node) || isSplit) {
|
|
// Expect non-associative operators to match exactly
|
|
var childMatches = [];
|
|
for (var i = 0; i < rule.args.length; i++) {
|
|
var childMatch = _ruleMatch(rule.args[i], node.args[i]);
|
|
if (childMatch.length === 0) {
|
|
// Child did not match, so stop searching immediately
|
|
return [];
|
|
}
|
|
// The child matched, so add the information returned from the child to our result
|
|
childMatches.push(childMatch);
|
|
}
|
|
res = mergeChildMatches(childMatches);
|
|
}
|
|
else if (node.args.length >= 2 && rule.args.length === 2) { // node is flattened, rule is not
|
|
// Associative operators/functions can be split in different ways so we check if the rule matches each
|
|
// them and return their union.
|
|
var splits = getSplits(node, rule.context);
|
|
var splitMatches = [];
|
|
for(var i = 0; i < splits.length; i++) {
|
|
var matchSet = _ruleMatch(rule, splits[i], true); // recursing at the same tree depth here
|
|
splitMatches = splitMatches.concat(matchSet);
|
|
}
|
|
return splitMatches;
|
|
}
|
|
else if (rule.args.length > 2) {
|
|
throw Error('Unexpected non-binary associative function: ' + rule.toString());
|
|
}
|
|
else {
|
|
// Incorrect number of arguments in rule and node, so no match
|
|
return [];
|
|
}
|
|
}
|
|
else if (rule instanceof SymbolNode) {
|
|
// If the rule is a SymbolNode, then it carries a special meaning
|
|
// according to the first character of the symbol node name.
|
|
// c.* matches a ConstantNode
|
|
// n.* matches any node
|
|
if (rule.name.length === 0) {
|
|
throw new Error('Symbol in rule has 0 length...!?');
|
|
}
|
|
if (math.hasOwnProperty(rule.name)) {
|
|
if (!SUPPORTED_CONSTANTS[rule.name]) {
|
|
throw new Error('Built in constant: ' + rule.name + ' is not supported by simplify.');
|
|
}
|
|
|
|
// built-in constant must match exactly
|
|
if(rule.name !== node.name) {
|
|
return [];
|
|
}
|
|
}
|
|
else if (rule.name[0] === 'n' || rule.name.substring(0,2) === '_p') {
|
|
// rule matches _anything_, so assign this node to the rule.name placeholder
|
|
// Assign node to the rule.name placeholder.
|
|
// Our parent will check for matches among placeholders.
|
|
res[0].placeholders[rule.name] = node;
|
|
}
|
|
else if (rule.name[0] === 'v') {
|
|
// rule matches any variable thing (not a ConstantNode)
|
|
if(!type.isConstantNode(node)) {
|
|
res[0].placeholders[rule.name] = node;
|
|
}
|
|
else {
|
|
// Mis-match: rule was expecting something other than a ConstantNode
|
|
return [];
|
|
}
|
|
}
|
|
else if (rule.name[0] === 'c') {
|
|
// rule matches any ConstantNode
|
|
if(node instanceof ConstantNode) {
|
|
res[0].placeholders[rule.name] = node;
|
|
}
|
|
else {
|
|
// Mis-match: rule was expecting a ConstantNode
|
|
return [];
|
|
}
|
|
}
|
|
else {
|
|
throw new Error('Invalid symbol in rule: ' + rule.name);
|
|
}
|
|
}
|
|
else if (rule instanceof ConstantNode) {
|
|
// Literal constant must match exactly
|
|
if(rule.value !== node.value) {
|
|
return [];
|
|
}
|
|
}
|
|
else {
|
|
// Some other node was encountered which we aren't prepared for, so no match
|
|
return [];
|
|
}
|
|
|
|
// It's a match!
|
|
|
|
// console.log('_ruleMatch(' + rule.toString() + ', ' + node.toString() + ') found a match');
|
|
return res;
|
|
}
|
|
|
|
|
|
/**
|
|
* Determines whether p and q (and all their children nodes) are identical.
|
|
*
|
|
* @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} p
|
|
* @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} q
|
|
* @return {Object} Information about the match, if it exists.
|
|
*/
|
|
function _exactMatch(p, q) {
|
|
if(p instanceof ConstantNode && q instanceof ConstantNode) {
|
|
if(p.value !== q.value) {
|
|
return false;
|
|
}
|
|
}
|
|
else if(p instanceof SymbolNode && q instanceof SymbolNode) {
|
|
if(p.name !== q.name) {
|
|
return false;
|
|
}
|
|
}
|
|
else if(p instanceof OperatorNode && q instanceof OperatorNode
|
|
|| p instanceof FunctionNode && q instanceof FunctionNode) {
|
|
if (p instanceof OperatorNode) {
|
|
if (p.op !== q.op || p.fn !== q.fn) {
|
|
return false;
|
|
}
|
|
}
|
|
else if (p instanceof FunctionNode) {
|
|
if (p.name !== q.name) {
|
|
return false;
|
|
}
|
|
}
|
|
|
|
if(p.args.length !== q.args.length) {
|
|
return false;
|
|
}
|
|
|
|
for(var i=0; i<p.args.length; i++) {
|
|
if(!_exactMatch(p.args[i], q.args[i])) {
|
|
return false;
|
|
}
|
|
}
|
|
}
|
|
else {
|
|
return false;
|
|
}
|
|
|
|
return true;
|
|
}
|
|
|
|
return simplify;
|
|
}
|
|
|
|
exports.math = true;
|
|
exports.name = 'simplify';
|
|
exports.factory = factory;
|