"use strict";
Object.defineProperty(exports, "__esModule", {
value: true
});
exports.createRationalize = void 0;
var _number = require("../../utils/number.js");
var _factory = require("../../utils/factory.js");
const name = 'rationalize';
const dependencies = ['config', 'typed', 'equal', 'isZero', 'add', 'subtract', 'multiply', 'divide', 'pow', 'parse', 'simplifyConstant', 'simplifyCore', 'simplify', '?bignumber', '?fraction', 'mathWithTransform', 'matrix', 'AccessorNode', 'ArrayNode', 'ConstantNode', 'FunctionNode', 'IndexNode', 'ObjectNode', 'OperatorNode', 'SymbolNode', 'ParenthesisNode'];
const createRationalize = exports.createRationalize = /* #__PURE__ */(0, _factory.factory)(name, dependencies, _ref => {
let {
config,
typed,
equal,
isZero,
add,
subtract,
multiply,
divide,
pow,
parse,
simplifyConstant,
simplifyCore,
simplify,
fraction,
bignumber,
mathWithTransform,
matrix,
AccessorNode,
ArrayNode,
ConstantNode,
FunctionNode,
IndexNode,
ObjectNode,
OperatorNode,
SymbolNode,
ParenthesisNode
} = _ref;
/**
* Transform a rationalizable expression in a rational fraction.
* If rational fraction is one variable polynomial then converts
* the numerator and denominator in canonical form, with decreasing
* exponents, returning the coefficients of numerator.
*
* Syntax:
*
* math.rationalize(expr)
* math.rationalize(expr, detailed)
* math.rationalize(expr, scope)
* math.rationalize(expr, scope, detailed)
*
* Examples:
*
* math.rationalize('sin(x)+y')
* // Error: There is an unsolved function call
* math.rationalize('2x/y - y/(x+1)')
* // (2*x^2-y^2+2*x)/(x*y+y)
* math.rationalize('(2x+1)^6')
* // 64*x^6+192*x^5+240*x^4+160*x^3+60*x^2+12*x+1
* math.rationalize('2x/( (2x-1) / (3x+2) ) - 5x/ ( (3x+4) / (2x^2-5) ) + 3')
* // -20*x^4+28*x^3+104*x^2+6*x-12)/(6*x^2+5*x-4)
* math.rationalize('x/(1-x)/(x-2)/(x-3)/(x-4) + 2x/ ( (1-2x)/(2-3x) )/ ((3-4x)/(4-5x) )') =
* // (-30*x^7+344*x^6-1506*x^5+3200*x^4-3472*x^3+1846*x^2-381*x)/
* // (-8*x^6+90*x^5-383*x^4+780*x^3-797*x^2+390*x-72)
*
* math.rationalize('x+x+x+y',{y:1}) // 3*x+1
* math.rationalize('x+x+x+y',{}) // 3*x+y
*
* const ret = math.rationalize('x+x+x+y',{},true)
* // ret.expression=3*x+y, ret.variables = ["x","y"]
* const ret = math.rationalize('-2+5x^2',{},true)
* // ret.expression=5*x^2-2, ret.variables = ["x"], ret.coefficients=[-2,0,5]
*
* See also:
*
* simplify
*
* @param {Node|string} expr The expression to check if is a polynomial expression
* @param {Object|boolean} optional scope of expression or true for already evaluated rational expression at input
* @param {Boolean} detailed optional True if return an object, false if return expression node (default)
*
* @return {Object | Node} The rational polynomial of `expr` or an object
* `{expression, numerator, denominator, variables, coefficients}`, where
* `expression` is a `Node` with the node simplified expression,
* `numerator` is a `Node` with the simplified numerator of expression,
* `denominator` is a `Node` or `boolean` with the simplified denominator or `false` (if there is no denominator),
* `variables` is an array with variable names,
* and `coefficients` is an array with coefficients of numerator sorted by increased exponent
* {Expression Node} node simplified expression
*
*/
function _rationalize(expr) {
let scope = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {};
let detailed = arguments.length > 2 && arguments[2] !== undefined ? arguments[2] : false;
const setRules = rulesRationalize(); // Rules for change polynomial in near canonical form
const polyRet = polynomial(expr, scope, true, setRules.firstRules); // Check if expression is a rationalizable polynomial
const nVars = polyRet.variables.length;
const noExactFractions = {
exactFractions: false
};
const withExactFractions = {
exactFractions: true
};
expr = polyRet.expression;
if (nVars >= 1) {
// If expression in not a constant
expr = expandPower(expr); // First expand power of polynomials (cannot be made from rules!)
let sBefore; // Previous expression
let rules;
let eDistrDiv = true;
let redoInic = false;
// Apply the initial rules, including succ div rules:
expr = simplify(expr, setRules.firstRules, {}, noExactFractions);
let s;
while (true) {
// Alternate applying successive division rules and distr.div.rules
// until there are no more changes:
rules = eDistrDiv ? setRules.distrDivRules : setRules.sucDivRules;
expr = simplify(expr, rules, {}, withExactFractions);
eDistrDiv = !eDistrDiv; // Swap between Distr.Div and Succ. Div. Rules
s = expr.toString();
if (s === sBefore) {
break; // No changes : end of the loop
}
redoInic = true;
sBefore = s;
}
if (redoInic) {
// Apply first rules again without succ div rules (if there are changes)
expr = simplify(expr, setRules.firstRulesAgain, {}, noExactFractions);
}
// Apply final rules:
expr = simplify(expr, setRules.finalRules, {}, noExactFractions);
} // NVars >= 1
const coefficients = [];
const retRationalize = {};
if (expr.type === 'OperatorNode' && expr.isBinary() && expr.op === '/') {
// Separate numerator from denominator
if (nVars === 1) {
expr.args[0] = polyToCanonical(expr.args[0], coefficients);
expr.args[1] = polyToCanonical(expr.args[1]);
}
if (detailed) {
retRationalize.numerator = expr.args[0];
retRationalize.denominator = expr.args[1];
}
} else {
if (nVars === 1) {
expr = polyToCanonical(expr, coefficients);
}
if (detailed) {
retRationalize.numerator = expr;
retRationalize.denominator = null;
}
}
// nVars
if (!detailed) return expr;
retRationalize.coefficients = coefficients;
retRationalize.variables = polyRet.variables;
retRationalize.expression = expr;
return retRationalize;
}
return typed(name, {
Node: _rationalize,
'Node, boolean': (expr, detailed) => _rationalize(expr, {}, detailed),
'Node, Object': _rationalize,
'Node, Object, boolean': _rationalize
}); // end of typed rationalize
/**
* Function to simplify an expression using an optional scope and
* return it if the expression is a polynomial expression, i.e.
* an expression with one or more variables and the operators
* +, -, *, and ^, where the exponent can only be a positive integer.
*
* Syntax:
*
* polynomial(expr,scope,extended, rules)
*
* @param {Node | string} expr The expression to simplify and check if is polynomial expression
* @param {object} scope Optional scope for expression simplification
* @param {boolean} extended Optional. Default is false. When true allows divide operator.
* @param {array} rules Optional. Default is no rule.
*
*
* @return {Object}
* {Object} node: node simplified expression
* {Array} variables: variable names
*/
function polynomial(expr, scope, extended, rules) {
const variables = [];
const node = simplify(expr, rules, scope, {
exactFractions: false
}); // Resolves any variables and functions with all defined parameters
extended = !!extended;
const oper = '+-*' + (extended ? '/' : '');
recPoly(node);
const retFunc = {};
retFunc.expression = node;
retFunc.variables = variables;
return retFunc;
// -------------------------------------------------------------------------------------------------------
/**
* Function to simplify an expression using an optional scope and
* return it if the expression is a polynomial expression, i.e.
* an expression with one or more variables and the operators
* +, -, *, and ^, where the exponent can only be a positive integer.
*
* Syntax:
*
* recPoly(node)
*
*
* @param {Node} node The current sub tree expression in recursion
*
* @return nothing, throw an exception if error
*/
function recPoly(node) {
const tp = node.type; // node type
if (tp === 'FunctionNode') {
// No function call in polynomial expression
throw new Error('There is an unsolved function call');
} else if (tp === 'OperatorNode') {
if (node.op === '^') {
// TODO: handle negative exponents like in '1/x^(-2)'
if (node.args[1].type !== 'ConstantNode' || !(0, _number.isInteger)(parseFloat(node.args[1].value))) {
throw new Error('There is a non-integer exponent');
} else {
recPoly(node.args[0]);
}
} else {
if (!oper.includes(node.op)) {
throw new Error('Operator ' + node.op + ' invalid in polynomial expression');
}
for (let i = 0; i < node.args.length; i++) {
recPoly(node.args[i]);
}
} // type of operator
} else if (tp === 'SymbolNode') {
const name = node.name; // variable name
const pos = variables.indexOf(name);
if (pos === -1) {
// new variable in expression
variables.push(name);
}
} else if (tp === 'ParenthesisNode') {
recPoly(node.content);
} else if (tp !== 'ConstantNode') {
throw new Error('type ' + tp + ' is not allowed in polynomial expression');
}
} // end of recPoly
} // end of polynomial
// ---------------------------------------------------------------------------------------
/**
* Return a rule set to rationalize an polynomial expression in rationalize
*
* Syntax:
*
* rulesRationalize()
*
* @return {array} rule set to rationalize an polynomial expression
*/
function rulesRationalize() {
const oldRules = [simplifyCore,
// sCore
{
l: 'n+n',
r: '2*n'
}, {
l: 'n+-n',
r: '0'
}, simplifyConstant,
// sConstant
{
l: 'n*(n1^-1)',
r: 'n/n1'
}, {
l: 'n*n1^-n2',
r: 'n/n1^n2'
}, {
l: 'n1^-1',
r: '1/n1'
}, {
l: 'n*(n1/n2)',
r: '(n*n1)/n2'
}, {
l: '1*n',
r: 'n'
}];
const rulesFirst = [{
l: '(-n1)/(-n2)',
r: 'n1/n2'
},
// Unary division
{
l: '(-n1)*(-n2)',
r: 'n1*n2'
},
// Unary multiplication
{
l: 'n1--n2',
r: 'n1+n2'
},
// '--' elimination
{
l: 'n1-n2',
r: 'n1+(-n2)'
},
// Subtraction turn into add with un�ry minus
{
l: '(n1+n2)*n3',
r: '(n1*n3 + n2*n3)'
},
// Distributive 1
{
l: 'n1*(n2+n3)',
r: '(n1*n2+n1*n3)'
},
// Distributive 2
{
l: 'c1*n + c2*n',
r: '(c1+c2)*n'
},
// Joining constants
{
l: 'c1*n + n',
r: '(c1+1)*n'
},
// Joining constants
{
l: 'c1*n - c2*n',
r: '(c1-c2)*n'
},
// Joining constants
{
l: 'c1*n - n',
r: '(c1-1)*n'
},
// Joining constants
{
l: 'v/c',
r: '(1/c)*v'
},
// variable/constant (new!)
{
l: 'v/-c',
r: '-(1/c)*v'
},
// variable/constant (new!)
{
l: '-v*-c',
r: 'c*v'
},
// Inversion constant and variable 1
{
l: '-v*c',
r: '-c*v'
},
// Inversion constant and variable 2
{
l: 'v*-c',
r: '-c*v'
},
// Inversion constant and variable 3
{
l: 'v*c',
r: 'c*v'
},
// Inversion constant and variable 4
{
l: '-(-n1*n2)',
r: '(n1*n2)'
},
// Unary propagation
{
l: '-(n1*n2)',
r: '(-n1*n2)'
},
// Unary propagation
{
l: '-(-n1+n2)',
r: '(n1-n2)'
},
// Unary propagation
{
l: '-(n1+n2)',
r: '(-n1-n2)'
},
// Unary propagation
{
l: '(n1^n2)^n3',
r: '(n1^(n2*n3))'
},
// Power to Power
{
l: '-(-n1/n2)',
r: '(n1/n2)'
},
// Division and Unary
{
l: '-(n1/n2)',
r: '(-n1/n2)'
}]; // Divisao and Unary
const rulesDistrDiv = [{
l: '(n1/n2 + n3/n4)',
r: '((n1*n4 + n3*n2)/(n2*n4))'
},
// Sum of fractions
{
l: '(n1/n2 + n3)',
r: '((n1 + n3*n2)/n2)'
},
// Sum fraction with number 1
{
l: '(n1 + n2/n3)',
r: '((n1*n3 + n2)/n3)'
}]; // Sum fraction with number 1
const rulesSucDiv = [{
l: '(n1/(n2/n3))',
r: '((n1*n3)/n2)'
},
// Division simplification
{
l: '(n1/n2/n3)',
r: '(n1/(n2*n3))'
}];
const setRules = {}; // rules set in 4 steps.
// All rules => infinite loop
// setRules.allRules =oldRules.concat(rulesFirst,rulesDistrDiv,rulesSucDiv)
setRules.firstRules = oldRules.concat(rulesFirst, rulesSucDiv); // First rule set
setRules.distrDivRules = rulesDistrDiv; // Just distr. div. rules
setRules.sucDivRules = rulesSucDiv; // Jus succ. div. rules
setRules.firstRulesAgain = oldRules.concat(rulesFirst); // Last rules set without succ. div.
// Division simplification
// Second rule set.
// There is no aggregate expression with parentesis, but the only variable can be scattered.
setRules.finalRules = [simplifyCore,
// simplify.rules[0]
{
l: 'n*-n',
r: '-n^2'
},
// Joining multiply with power 1
{
l: 'n*n',
r: 'n^2'
},
// Joining multiply with power 2
simplifyConstant,
// simplify.rules[14] old 3rd index in oldRules
{
l: 'n*-n^n1',
r: '-n^(n1+1)'
},
// Joining multiply with power 3
{
l: 'n*n^n1',
r: 'n^(n1+1)'
},
// Joining multiply with power 4
{
l: 'n^n1*-n^n2',
r: '-n^(n1+n2)'
},
// Joining multiply with power 5
{
l: 'n^n1*n^n2',
r: 'n^(n1+n2)'
},
// Joining multiply with power 6
{
l: 'n^n1*-n',
r: '-n^(n1+1)'
},
// Joining multiply with power 7
{
l: 'n^n1*n',
r: 'n^(n1+1)'
},
// Joining multiply with power 8
{
l: 'n^n1/-n',
r: '-n^(n1-1)'
},
// Joining multiply with power 8
{
l: 'n^n1/n',
r: 'n^(n1-1)'
},
// Joining division with power 1
{
l: 'n/-n^n1',
r: '-n^(1-n1)'
},
// Joining division with power 2
{
l: 'n/n^n1',
r: 'n^(1-n1)'
},
// Joining division with power 3
{
l: 'n^n1/-n^n2',
r: 'n^(n1-n2)'
},
// Joining division with power 4
{
l: 'n^n1/n^n2',
r: 'n^(n1-n2)'
},
// Joining division with power 5
{
l: 'n1+(-n2*n3)',
r: 'n1-n2*n3'
},
// Solving useless parenthesis 1
{
l: 'v*(-c)',
r: '-c*v'
},
// Solving useless unary 2
{
l: 'n1+-n2',
r: 'n1-n2'
},
// Solving +- together (new!)
{
l: 'v*c',
r: 'c*v'
},
// inversion constant with variable
{
l: '(n1^n2)^n3',
r: '(n1^(n2*n3))'
} // Power to Power
];
return setRules;
} // End rulesRationalize
// ---------------------------------------------------------------------------------------
/**
* Expand recursively a tree node for handling with expressions with exponents
* (it's not for constants, symbols or functions with exponents)
* PS: The other parameters are internal for recursion
*
* Syntax:
*
* expandPower(node)
*
* @param {Node} node Current expression node
* @param {node} parent Parent current node inside the recursion
* @param (int} Parent number of chid inside the rercursion
*
* @return {node} node expression with all powers expanded.
*/
function expandPower(node, parent, indParent) {
const tp = node.type;
const internal = arguments.length > 1; // TRUE in internal calls
if (tp === 'OperatorNode' && node.isBinary()) {
let does = false;
let val;
if (node.op === '^') {
// First operator: Parenthesis or UnaryMinus
if ((node.args[0].type === 'ParenthesisNode' || node.args[0].type === 'OperatorNode') && node.args[1].type === 'ConstantNode') {
// Second operator: Constant
val = parseFloat(node.args[1].value);
does = val >= 2 && (0, _number.isInteger)(val);
}
}
if (does) {
// Exponent >= 2
// Before:
// operator A --> Subtree
// parent pow
// constant
//
if (val > 2) {
// Exponent > 2,
// AFTER: (exponent > 2)
// operator A --> Subtree
// parent *
// deep clone (operator A --> Subtree
// pow
// constant - 1
//
const nEsqTopo = node.args[0];
const nDirTopo = new OperatorNode('^', 'pow', [node.args[0].cloneDeep(), new ConstantNode(val - 1)]);
node = new OperatorNode('*', 'multiply', [nEsqTopo, nDirTopo]);
} else {
// Expo = 2 - no power
// AFTER: (exponent = 2)
// operator A --> Subtree
// parent oper
// deep clone (operator A --> Subtree)
//
node = new OperatorNode('*', 'multiply', [node.args[0], node.args[0].cloneDeep()]);
}
if (internal) {
// Change parent references in internal recursive calls
if (indParent === 'content') {
parent.content = node;
} else {
parent.args[indParent] = node;
}
}
} // does
} // binary OperatorNode
if (tp === 'ParenthesisNode') {
// Recursion
expandPower(node.content, node, 'content');
} else if (tp !== 'ConstantNode' && tp !== 'SymbolNode') {
for (let i = 0; i < node.args.length; i++) {
expandPower(node.args[i], node, i);
}
}
if (!internal) {
// return the root node
return node;
}
} // End expandPower
// ---------------------------------------------------------------------------------------
/**
* Auxilary function for rationalize
* Convert near canonical polynomial in one variable in a canonical polynomial
* with one term for each exponent in decreasing order
*
* Syntax:
*
* polyToCanonical(node [, coefficients])
*
* @param {Node | string} expr The near canonical polynomial expression to convert in a a canonical polynomial expression
*
* The string or tree expression needs to be at below syntax, with free spaces:
* ( (^(-)? | [+-]? )cte (*)? var (^expo)? | cte )+
* Where 'var' is one variable with any valid name
* 'cte' are real numeric constants with any value. It can be omitted if equal than 1
* 'expo' are integers greater than 0. It can be omitted if equal than 1.
*
* @param {array} coefficients Optional returns coefficients sorted by increased exponent
*
*
* @return {node} new node tree with one variable polynomial or string error.
*/
function polyToCanonical(node, coefficients) {
if (coefficients === undefined) {
coefficients = [];
} // coefficients.
coefficients[0] = 0; // index is the exponent
const o = {};
o.cte = 1;
o.oper = '+';
// fire: mark with * or ^ when finds * or ^ down tree, reset to "" with + and -.
// It is used to deduce the exponent: 1 for *, 0 for "".
o.fire = '';
let maxExpo = 0; // maximum exponent
let varname = ''; // variable name
recurPol(node, null, o);
maxExpo = coefficients.length - 1;
let first = true;
let no;
for (let i = maxExpo; i >= 0; i--) {
if (coefficients[i] === 0) continue;
let n1 = new ConstantNode(first ? coefficients[i] : Math.abs(coefficients[i]));
const op = coefficients[i] < 0 ? '-' : '+';
if (i > 0) {
// Is not a constant without variable
let n2 = new SymbolNode(varname);
if (i > 1) {
const n3 = new ConstantNode(i);
n2 = new OperatorNode('^', 'pow', [n2, n3]);
}
if (coefficients[i] === -1 && first) {
n1 = new OperatorNode('-', 'unaryMinus', [n2]);
} else if (Math.abs(coefficients[i]) === 1) {
n1 = n2;
} else {
n1 = new OperatorNode('*', 'multiply', [n1, n2]);
}
}
if (first) {
no = n1;
} else if (op === '+') {
no = new OperatorNode('+', 'add', [no, n1]);
} else {
no = new OperatorNode('-', 'subtract', [no, n1]);
}
first = false;
} // for
if (first) {
return new ConstantNode(0);
} else {
return no;
}
/**
* Recursive auxilary function inside polyToCanonical for
* converting expression in canonical form
*
* Syntax:
*
* recurPol(node, noPai, obj)
*
* @param {Node} node The current subpolynomial expression
* @param {Node | Null} noPai The current parent node
* @param {object} obj Object with many internal flags
*
* @return {} No return. If error, throws an exception
*/
function recurPol(node, noPai, o) {
const tp = node.type;
if (tp === 'FunctionNode') {
// ***** FunctionName *****
// No function call in polynomial expression
throw new Error('There is an unsolved function call');
} else if (tp === 'OperatorNode') {
// ***** OperatorName *****
if (!'+-*^'.includes(node.op)) throw new Error('Operator ' + node.op + ' invalid');
if (noPai !== null) {
// -(unary),^ : children of *,+,-
if ((node.fn === 'unaryMinus' || node.fn === 'pow') && noPai.fn !== 'add' && noPai.fn !== 'subtract' && noPai.fn !== 'multiply') {
throw new Error('Invalid ' + node.op + ' placing');
}
// -,+,* : children of +,-
if ((node.fn === 'subtract' || node.fn === 'add' || node.fn === 'multiply') && noPai.fn !== 'add' && noPai.fn !== 'subtract') {
throw new Error('Invalid ' + node.op + ' placing');
}
// -,+ : first child
if ((node.fn === 'subtract' || node.fn === 'add' || node.fn === 'unaryMinus') && o.noFil !== 0) {
throw new Error('Invalid ' + node.op + ' placing');
}
} // Has parent
// Firers: ^,* Old: ^,&,-(unary): firers
if (node.op === '^' || node.op === '*') {
o.fire = node.op;
}
for (let i = 0; i < node.args.length; i++) {
// +,-: reset fire
if (node.fn === 'unaryMinus') o.oper = '-';
if (node.op === '+' || node.fn === 'subtract') {
o.fire = '';
o.cte = 1; // default if there is no constant
o.oper = i === 0 ? '+' : node.op;
}
o.noFil = i; // number of son
recurPol(node.args[i], node, o);
} // for in children
} else if (tp === 'SymbolNode') {
// ***** SymbolName *****
if (node.name !== varname && varname !== '') {
throw new Error('There is more than one variable');
}
varname = node.name;
if (noPai === null) {
coefficients[1] = 1;
return;
}
// ^: Symbol is First child
if (noPai.op === '^' && o.noFil !== 0) {
throw new Error('In power the variable should be the first parameter');
}
// *: Symbol is Second child
if (noPai.op === '*' && o.noFil !== 1) {
throw new Error('In multiply the variable should be the second parameter');
}
// Symbol: firers '',* => it means there is no exponent above, so it's 1 (cte * var)
if (o.fire === '' || o.fire === '*') {
if (maxExpo < 1) coefficients[1] = 0;
coefficients[1] += o.cte * (o.oper === '+' ? 1 : -1);
maxExpo = Math.max(1, maxExpo);
}
} else if (tp === 'ConstantNode') {
const valor = parseFloat(node.value);
if (noPai === null) {
coefficients[0] = valor;
return;
}
if (noPai.op === '^') {
// cte: second child of power
if (o.noFil !== 1) throw new Error('Constant cannot be powered');
if (!(0, _number.isInteger)(valor) || valor <= 0) {
throw new Error('Non-integer exponent is not allowed');
}
for (let i = maxExpo + 1; i < valor; i++) coefficients[i] = 0;
if (valor > maxExpo) coefficients[valor] = 0;
coefficients[valor] += o.cte * (o.oper === '+' ? 1 : -1);
maxExpo = Math.max(valor, maxExpo);
return;
}
o.cte = valor;
// Cte: firer '' => There is no exponent and no multiplication, so the exponent is 0.
if (o.fire === '') {
coefficients[0] += o.cte * (o.oper === '+' ? 1 : -1);
}
} else {
throw new Error('Type ' + tp + ' is not allowed');
}
} // End of recurPol
} // End of polyToCanonical
});