将 LaTeX 转换为动态 Javascript 函数
我有一个方程的用户输入 - 这个输入使用我没有编码的单独 API 生成 LaTeX 代码(即，Mathquill，没关系).
I have a user input for an equation - this input generates LaTeX code using a separate API which I did not code (namely, Mathquill, not that it matters).
我的问题最好用一个例子来说明:假设从用户输入生成的 LaTeX 代码是这样的:
My problem is best illustrated by an example: suppose the LaTeX code generated from the user input was this:
x^2+3x-10sinleft(2xight)
我如何将它(当然是在运行中)转换成一个硬编码的 JavaScript 函数，如下所示:
How would I convert this (on the fly of course) into a JavaScript function which, hard-coded, would look like this:
function(x) {
return Math.pow(x, 2) + 3 * x - 10 * Math.sin(2 * x);
}
是否有任何 API，或者我是否正在考虑编写一些可以解释 LaTeX 符号并以某种方式生成函数的东西?还是什么?
Are there any APIs or am I looking at writing something which will interpret the LaTeX symbols and make a function, somehow? Or what?
推荐答案
我已经编写了一个(绝不是通用的)解决方案，主要基于 George 的代码.
I have written a (by no means general purpose) solution, heavily based on George's code.
这里是:
var CALC_CONST = {
// define your constants
e: Math.E,
pi: Math.PI
};
var CALC_NUMARGS = [
[/^(^|*|/|+|-)$/, 2],
[/^(floor|ceil|(sin|cos|tan|sec|csc|cot)h?)$/, 1]
];
var Calc = function(expr, infix) {
this.valid = true;
this.expr = expr;
if (!infix) {
// by default treat expr as raw latex
this.expr = this.latexToInfix(expr);
}
var OpPrecedence = function(op) {
if (typeof op == "undefined") return 0;
return op.match(/^(floor|ceil|(sin|cos|tan|sec|csc|cot)h?)$/) ? 10
: (op === "^") ? 9
: (op === "*" || op === "/") ? 8
: (op === "+" || op === "-") ? 7
: 0;
}
var OpAssociativity = function(op) {
return op.match(/^(floor|ceil|(sin|cos|tan|sec|csc|cot)h?)$/) ? "R" : "L";
}
var numArgs = function(op) {
for (var i = 0; i < CALC_NUMARGS.length; i++) {
if (CALC_NUMARGS[i][0].test(op)) return CALC_NUMARGS[i][1];
}
return false;
}
this.rpn_expr = [];
var rpn_expr = this.rpn_expr;
this.expr = this.expr.replace(/s+/g, "");
// This nice long regex matches any valid token in a user
// supplied expression (e.g. an operator, a constant or
// a variable)
var in_tokens = this.expr.match(/(^|*|/|+|-|(|)|[a-zA-Z0-9.]+)/gi);
var op_stack = [];
in_tokens.forEach(function(token) {
if (/^[a-zA-Z]$/.test(token)) {
if (CALC_CONST.hasOwnProperty(token)) {
// Constant. Pushes a value onto the stack.
rpn_expr.push(["num", CALC_CONST[token]]);
}
else {
// Variables (i.e. x as in f(x))
rpn_expr.push(["var", token]);
}
}
else {
var numVal = parseFloat(token);
if (!isNaN(numVal)) {
// Number - push onto the stack
rpn_expr.push(["num", numVal]);
}
else if (token === ")") {
// Pop tokens off the op_stack onto the rpn_expr until we reach the matching (
while (op_stack[op_stack.length - 1] !== "(") {
rpn_expr.push([numArgs(op_stack[op_stack.length - 1]), op_stack.pop()]);
if (op_stack.length === 0) {
this.valid = false;
return;
}
}
// remove the (
op_stack.pop();
}
else if (token === "(") {
op_stack.push(token);
}
else {
// Operator
var tokPrec = OpPrecedence(token),
headPrec = OpPrecedence(op_stack[op_stack.length - 1]);
while ((OpAssociativity(token) === "L" && tokPrec <= headPrec) ||
(OpAssociativity(token) === "R" && tokPrec < headPrec)) {
rpn_expr.push([numArgs(op_stack[op_stack.length - 1]), op_stack.pop()]);
if (op_stack.length === 0) break;
headPrec = OpPrecedence(op_stack[op_stack.length - 1]);
}
op_stack.push(token);
}
}
});
// Push all remaining operators onto the final expression
while (op_stack.length > 0) {
var popped = op_stack.pop();
if (popped === ")") {
this.valid = false;
break;
}
rpn_expr.push([numArgs(popped), popped]);
}
}
/**
* returns the result of evaluating the current expression
*/
Calc.prototype.eval = function(x) {
var stack = [], rpn_expr = this.rpn_expr;
rpn_expr.forEach(function(token) {
if (typeof token[0] == "string") {
switch (token[0]) {
case "var":
// Variable, i.e. x as in f(x); push value onto stack
//if (token[1] != "x") return false;
stack.push(x);
break;
case "num":
// Number; push value onto stack
stack.push(token[1]);
break;
}
}
else {
// Operator
var numArgs = token[0];
var args = [];
do {
args.unshift(stack.pop());
} while (args.length < numArgs);
switch (token[1]) {
/* BASIC ARITHMETIC OPERATORS */
case "*":
stack.push(args[0] * args[1]);
break;
case "/":
stack.push(args[0] / args[1]);
break;
case "+":
stack.push(args[0] + args[1]);
break;
case "-":
stack.push(args[0] - args[1]);
break;
// exponents
case "^":
stack.push(Math.pow(args[0], args[1]));
break;
/* TRIG FUNCTIONS */
case "sin":
stack.push(Math.sin(args[0]));
break;
case "cos":
stack.push(Math.cos(args[0]));
break;
case "tan":
stack.push(Math.tan(args[0]));
break;
case "sec":
stack.push(1 / Math.cos(args[0]));
break;
case "csc":
stack.push(1 / Math.sin(args[0]));
break;
case "cot":
stack.push(1 / Math.tan(args[0]));
break;
case "sinh":
stack.push(.5 * (Math.pow(Math.E, args[0]) - Math.pow(Math.E, -args[0])));
break;
case "cosh":
stack.push(.5 * (Math.pow(Math.E, args[0]) + Math.pow(Math.E, -args[0])));
break;
case "tanh":
stack.push((Math.pow(Math.E, 2*args[0]) - 1) / (Math.pow(Math.E, 2*args[0]) + 1));
break;
case "sech":
stack.push(2 / (Math.pow(Math.E, args[0]) + Math.pow(Math.E, -args[0])));
break;
case "csch":
stack.push(2 / (Math.pow(Math.E, args[0]) - Math.pow(Math.E, -args[0])));
break;
case "coth":
stack.push((Math.pow(Math.E, 2*args[0]) + 1) / (Math.pow(Math.E, 2*args[0]) - 1));
break;
case "floor":
stack.push(Math.floor(args[0]));
break;
case "ceil":
stack.push(Math.ceil(args[0]));
break;
default:
// unknown operator; error out
return false;
}
}
});
return stack.pop();
};
Calc.prototype.latexToInfix = function(latex) {
/**
* function: converts latex notation to infix notation (human-readable, to be converted
* again to prefix in order to be processed
*
* Supported functions / operators / notation:
* parentheses, exponents, adding, subtracting, multipling, dividing, fractions
* trigonometric (including hyperbolic) functions, floor, ceil
*/
var infix = latex;
infix = infix
.replace(/\frac{([^}]+)}{([^}]+)}/g, "($1)/($2)") // fractions
.replace(/\left(/g, "(") // open parenthesis
.replace(/\right)/g, ")") // close parenthesis
.replace(/[^(](floor|ceil|(sin|cos|tan|sec|csc|cot)h?)(([^()]+))[^)]/g, "($&)") // functions
.replace(/([^(floor|ceil|(sin|cos|tan|sec|csc|cot)h?|+|-|*|/)])(/g, "$1*(")
.replace(/)([w])/g, ")*$1")
.replace(/([0-9])([A-Za-z])/g, "$1*$2")
;
return infix;
};
使用示例:
var latex = "e^x+\frac{2}{3}x-4sin\left(x\right)";
var calc = new Calc(latex);
var test = calc.eval(3.5); // 36.85191820278412
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