ysjj-system/node_modules/mathjs/lib/esm/function/algebra/derivative.js

import { isConstantNode, typeOf } from '../../utils/is.js';
import { factory } from '../../utils/factory.js';
var name = 'derivative';
var dependencies = ['typed', 'config', 'parse', 'simplify', 'equal', 'isZero', 'numeric', 'ConstantNode', 'FunctionNode', 'OperatorNode', 'ParenthesisNode', 'SymbolNode'];
export var createDerivative = /* #__PURE__ */factory(name, dependencies, _ref => {
  var {
    typed,
    config,
    parse,
    simplify,
    equal,
    isZero,
    numeric,
    ConstantNode,
    FunctionNode,
    OperatorNode,
    ParenthesisNode,
    SymbolNode
  } = _ref;
  /**
   * Takes the derivative of an expression expressed in parser Nodes.
   * The derivative will be taken over the supplied variable in the
   * second parameter. If there are multiple variables in the expression,
   * it will return a partial derivative.
   *
   * This uses rules of differentiation which can be found here:
   *
   * - [Differentiation rules (Wikipedia)](https://en.wikipedia.org/wiki/Differentiation_rules)
   *
   * Syntax:
   *
   *     derivative(expr, variable)
   *     derivative(expr, variable, options)
   *
   * Examples:
   *
   *     math.derivative('x^2', 'x')                     // Node '2 * x'
   *     math.derivative('x^2', 'x', {simplify: false})  // Node '2 * 1 * x ^ (2 - 1)'
   *     math.derivative('sin(2x)', 'x'))                // Node '2 * cos(2 * x)'
   *     math.derivative('2*x', 'x').evaluate()          // number 2
   *     math.derivative('x^2', 'x').evaluate({x: 4})    // number 8
   *     const f = math.parse('x^2')
   *     const x = math.parse('x')
   *     math.derivative(f, x)                           // Node {2 * x}
   *
   * See also:
   *
   *     simplify, parse, evaluate
   *
   * @param  {Node | string} expr           The expression to differentiate
   * @param  {SymbolNode | string} variable The variable over which to differentiate
   * @param  {{simplify: boolean}} [options]
   *                         There is one option available, `simplify`, which
   *                         is true by default. When false, output will not
   *                         be simplified.
   * @return {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode}    The derivative of `expr`
   */
  function plainDerivative(expr, variable) {
    var options = arguments.length > 2 && arguments[2] !== undefined ? arguments[2] : {
      simplify: true
    };
    var constNodes = {};
    constTag(constNodes, expr, variable.name);
    var res = _derivative(expr, constNodes);
    return options.simplify ? simplify(res) : res;
  }
  typed.addConversion({
    from: 'identifier',
    to: 'SymbolNode',
    convert: parse
  });
  var derivative = typed(name, {
    'Node, SymbolNode': plainDerivative,
    'Node, SymbolNode, Object': plainDerivative

    /* TODO: implement and test syntax with order of derivatives -> implement as an option {order: number}
    'Node, SymbolNode, ConstantNode': function (expr, variable, {order}) {
      let res = expr
      for (let i = 0; i < order; i++) {
        let constNodes = {}
        constTag(constNodes, expr, variable.name)
        res = _derivative(res, constNodes)
      }
      return res
    }
    */
  });

  typed.removeConversion({
    from: 'identifier',
    to: 'SymbolNode',
    convert: parse
  });
  derivative._simplify = true;
  derivative.toTex = function (deriv) {
    return _derivTex.apply(null, deriv.args);
  };

  // FIXME: move the toTex method of derivative to latex.js. Difficulty is that it relies on parse.
  // NOTE: the optional "order" parameter here is currently unused
  var _derivTex = typed('_derivTex', {
    'Node, SymbolNode': function NodeSymbolNode(expr, x) {
      if (isConstantNode(expr) && typeOf(expr.value) === 'string') {
        return _derivTex(parse(expr.value).toString(), x.toString(), 1);
      } else {
        return _derivTex(expr.toTex(), x.toString(), 1);
      }
    },
    'Node, ConstantNode': function NodeConstantNode(expr, x) {
      if (typeOf(x.value) === 'string') {
        return _derivTex(expr, parse(x.value));
      } else {
        throw new Error("The second parameter to 'derivative' is a non-string constant");
      }
    },
    'Node, SymbolNode, ConstantNode': function NodeSymbolNodeConstantNode(expr, x, order) {
      return _derivTex(expr.toString(), x.name, order.value);
    },
    'string, string, number': function stringStringNumber(expr, x, order) {
      var d;
      if (order === 1) {
        d = '{d\\over d' + x + '}';
      } else {
        d = '{d^{' + order + '}\\over d' + x + '^{' + order + '}}';
      }
      return d + "\\left[".concat(expr, "\\right]");
    }
  });

  /**
   * Does a depth-first search on the expression tree to identify what Nodes
   * are constants (e.g. 2 + 2), and stores the ones that are constants in
   * constNodes. Classification is done as follows:
   *
   *   1. ConstantNodes are constants.
   *   2. If there exists a SymbolNode, of which we are differentiating over,
   *      in the subtree it is not constant.
   *
   * @param  {Object} constNodes  Holds the nodes that are constant
   * @param  {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
   * @param  {string} varName     Variable that we are differentiating
   * @return {boolean}  if node is constant
   */
  // TODO: can we rewrite constTag into a pure function?
  var constTag = typed('constTag', {
    'Object, ConstantNode, string': function ObjectConstantNodeString(constNodes, node) {
      constNodes[node] = true;
      return true;
    },
    'Object, SymbolNode, string': function ObjectSymbolNodeString(constNodes, node, varName) {
      // Treat other variables like constants. For reasoning, see:
      //   https://en.wikipedia.org/wiki/Partial_derivative
      if (node.name !== varName) {
        constNodes[node] = true;
        return true;
      }
      return false;
    },
    'Object, ParenthesisNode, string': function ObjectParenthesisNodeString(constNodes, node, varName) {
      return constTag(constNodes, node.content, varName);
    },
    'Object, FunctionAssignmentNode, string': function ObjectFunctionAssignmentNodeString(constNodes, node, varName) {
      if (node.params.indexOf(varName) === -1) {
        constNodes[node] = true;
        return true;
      }
      return constTag(constNodes, node.expr, varName);
    },
    'Object, FunctionNode | OperatorNode, string': function ObjectFunctionNodeOperatorNodeString(constNodes, node, varName) {
      if (node.args.length > 0) {
        var isConst = constTag(constNodes, node.args[0], varName);
        for (var i = 1; i < node.args.length; ++i) {
          isConst = constTag(constNodes, node.args[i], varName) && isConst;
        }
        if (isConst) {
          constNodes[node] = true;
          return true;
        }
      }
      return false;
    }
  });

  /**
   * Applies differentiation rules.
   *
   * @param  {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
   * @param  {Object} constNodes  Holds the nodes that are constant
   * @return {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode}    The derivative of `expr`
   */
  var _derivative = typed('_derivative', {
    'ConstantNode, Object': function ConstantNodeObject(node) {
      return createConstantNode(0);
    },
    'SymbolNode, Object': function SymbolNodeObject(node, constNodes) {
      if (constNodes[node] !== undefined) {
        return createConstantNode(0);
      }
      return createConstantNode(1);
    },
    'ParenthesisNode, Object': function ParenthesisNodeObject(node, constNodes) {
      return new ParenthesisNode(_derivative(node.content, constNodes));
    },
    'FunctionAssignmentNode, Object': function FunctionAssignmentNodeObject(node, constNodes) {
      if (constNodes[node] !== undefined) {
        return createConstantNode(0);
      }
      return _derivative(node.expr, constNodes);
    },
    'FunctionNode, Object': function FunctionNodeObject(node, constNodes) {
      if (node.args.length !== 1) {
        funcArgsCheck(node);
      }
      if (constNodes[node] !== undefined) {
        return createConstantNode(0);
      }
      var arg0 = node.args[0];
      var arg1;
      var div = false; // is output a fraction?
      var negative = false; // is output negative?

      var funcDerivative;
      switch (node.name) {
        case 'cbrt':
          // d/dx(cbrt(x)) = 1 / (3x^(2/3))
          div = true;
          funcDerivative = new OperatorNode('*', 'multiply', [createConstantNode(3), new OperatorNode('^', 'pow', [arg0, new OperatorNode('/', 'divide', [createConstantNode(2), createConstantNode(3)])])]);
          break;
        case 'sqrt':
        case 'nthRoot':
          // d/dx(sqrt(x)) = 1 / (2*sqrt(x))
          if (node.args.length === 1) {
            div = true;
            funcDerivative = new OperatorNode('*', 'multiply', [createConstantNode(2), new FunctionNode('sqrt', [arg0])]);
          } else if (node.args.length === 2) {
            // Rearrange from nthRoot(x, a) -> x^(1/a)
            arg1 = new OperatorNode('/', 'divide', [createConstantNode(1), node.args[1]]);

            // Is a variable?
            constNodes[arg1] = constNodes[node.args[1]];
            return _derivative(new OperatorNode('^', 'pow', [arg0, arg1]), constNodes);
          }
          break;
        case 'log10':
          arg1 = createConstantNode(10);
        /* fall through! */
        case 'log':
          if (!arg1 && node.args.length === 1) {
            // d/dx(log(x)) = 1 / x
            funcDerivative = arg0.clone();
            div = true;
          } else if (node.args.length === 1 && arg1 || node.args.length === 2 && constNodes[node.args[1]] !== undefined) {
            // d/dx(log(x, c)) = 1 / (x*ln(c))
            funcDerivative = new OperatorNode('*', 'multiply', [arg0.clone(), new FunctionNode('log', [arg1 || node.args[1]])]);
            div = true;
          } else if (node.args.length === 2) {
            // d/dx(log(f(x), g(x))) = d/dx(log(f(x)) / log(g(x)))
            return _derivative(new OperatorNode('/', 'divide', [new FunctionNode('log', [arg0]), new FunctionNode('log', [node.args[1]])]), constNodes);
          }
          break;
        case 'pow':
          constNodes[arg1] = constNodes[node.args[1]];
          // Pass to pow operator node parser
          return _derivative(new OperatorNode('^', 'pow', [arg0, node.args[1]]), constNodes);
        case 'exp':
          // d/dx(e^x) = e^x
          funcDerivative = new FunctionNode('exp', [arg0.clone()]);
          break;
        case 'sin':
          // d/dx(sin(x)) = cos(x)
          funcDerivative = new FunctionNode('cos', [arg0.clone()]);
          break;
        case 'cos':
          // d/dx(cos(x)) = -sin(x)
          funcDerivative = new OperatorNode('-', 'unaryMinus', [new FunctionNode('sin', [arg0.clone()])]);
          break;
        case 'tan':
          // d/dx(tan(x)) = sec(x)^2
          funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('sec', [arg0.clone()]), createConstantNode(2)]);
          break;
        case 'sec':
          // d/dx(sec(x)) = sec(x)tan(x)
          funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('tan', [arg0.clone()])]);
          break;
        case 'csc':
          // d/dx(csc(x)) = -csc(x)cot(x)
          negative = true;
          funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('cot', [arg0.clone()])]);
          break;
        case 'cot':
          // d/dx(cot(x)) = -csc(x)^2
          negative = true;
          funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('csc', [arg0.clone()]), createConstantNode(2)]);
          break;
        case 'asin':
          // d/dx(asin(x)) = 1 / sqrt(1 - x^2)
          div = true;
          funcDerivative = new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])])]);
          break;
        case 'acos':
          // d/dx(acos(x)) = -1 / sqrt(1 - x^2)
          div = true;
          negative = true;
          funcDerivative = new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])])]);
          break;
        case 'atan':
          // d/dx(atan(x)) = 1 / (x^2 + 1)
          div = true;
          funcDerivative = new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)]);
          break;
        case 'asec':
          // d/dx(asec(x)) = 1 / (|x|*sqrt(x^2 - 1))
          div = true;
          funcDerivative = new OperatorNode('*', 'multiply', [new FunctionNode('abs', [arg0.clone()]), new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])])]);
          break;
        case 'acsc':
          // d/dx(acsc(x)) = -1 / (|x|*sqrt(x^2 - 1))
          div = true;
          negative = true;
          funcDerivative = new OperatorNode('*', 'multiply', [new FunctionNode('abs', [arg0.clone()]), new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])])]);
          break;
        case 'acot':
          // d/dx(acot(x)) = -1 / (x^2 + 1)
          div = true;
          negative = true;
          funcDerivative = new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)]);
          break;
        case 'sinh':
          // d/dx(sinh(x)) = cosh(x)
          funcDerivative = new FunctionNode('cosh', [arg0.clone()]);
          break;
        case 'cosh':
          // d/dx(cosh(x)) = sinh(x)
          funcDerivative = new FunctionNode('sinh', [arg0.clone()]);
          break;
        case 'tanh':
          // d/dx(tanh(x)) = sech(x)^2
          funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('sech', [arg0.clone()]), createConstantNode(2)]);
          break;
        case 'sech':
          // d/dx(sech(x)) = -sech(x)tanh(x)
          negative = true;
          funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('tanh', [arg0.clone()])]);
          break;
        case 'csch':
          // d/dx(csch(x)) = -csch(x)coth(x)
          negative = true;
          funcDerivative = new OperatorNode('*', 'multiply', [node, new FunctionNode('coth', [arg0.clone()])]);
          break;
        case 'coth':
          // d/dx(coth(x)) = -csch(x)^2
          negative = true;
          funcDerivative = new OperatorNode('^', 'pow', [new FunctionNode('csch', [arg0.clone()]), createConstantNode(2)]);
          break;
        case 'asinh':
          // d/dx(asinh(x)) = 1 / sqrt(x^2 + 1)
          div = true;
          funcDerivative = new FunctionNode('sqrt', [new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])]);
          break;
        case 'acosh':
          // d/dx(acosh(x)) = 1 / sqrt(x^2 - 1); XXX potentially only for x >= 1 (the real spectrum)
          div = true;
          funcDerivative = new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])]);
          break;
        case 'atanh':
          // d/dx(atanh(x)) = 1 / (1 - x^2)
          div = true;
          funcDerivative = new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])]);
          break;
        case 'asech':
          // d/dx(asech(x)) = -1 / (x*sqrt(1 - x^2))
          div = true;
          negative = true;
          funcDerivative = new OperatorNode('*', 'multiply', [arg0.clone(), new FunctionNode('sqrt', [new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])])])]);
          break;
        case 'acsch':
          // d/dx(acsch(x)) = -1 / (|x|*sqrt(x^2 + 1))
          div = true;
          negative = true;
          funcDerivative = new OperatorNode('*', 'multiply', [new FunctionNode('abs', [arg0.clone()]), new FunctionNode('sqrt', [new OperatorNode('+', 'add', [new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)]), createConstantNode(1)])])]);
          break;
        case 'acoth':
          // d/dx(acoth(x)) = -1 / (1 - x^2)
          div = true;
          negative = true;
          funcDerivative = new OperatorNode('-', 'subtract', [createConstantNode(1), new OperatorNode('^', 'pow', [arg0.clone(), createConstantNode(2)])]);
          break;
        case 'abs':
          // d/dx(abs(x)) = abs(x)/x
          funcDerivative = new OperatorNode('/', 'divide', [new FunctionNode(new SymbolNode('abs'), [arg0.clone()]), arg0.clone()]);
          break;
        case 'gamma': // Needs digamma function, d/dx(gamma(x)) = gamma(x)digamma(x)
        default:
          throw new Error('Function "' + node.name + '" is not supported by derivative, or a wrong number of arguments is passed');
      }
      var op, func;
      if (div) {
        op = '/';
        func = 'divide';
      } else {
        op = '*';
        func = 'multiply';
      }

      /* Apply chain rule to all functions:
         F(x)  = f(g(x))
         F'(x) = g'(x)*f'(g(x)) */
      var chainDerivative = _derivative(arg0, constNodes);
      if (negative) {
        chainDerivative = new OperatorNode('-', 'unaryMinus', [chainDerivative]);
      }
      return new OperatorNode(op, func, [chainDerivative, funcDerivative]);
    },
    'OperatorNode, Object': function OperatorNodeObject(node, constNodes) {
      if (constNodes[node] !== undefined) {
        return createConstantNode(0);
      }
      if (node.op === '+') {
        // d/dx(sum(f(x)) = sum(f'(x))
        return new OperatorNode(node.op, node.fn, node.args.map(function (arg) {
          return _derivative(arg, constNodes);
        }));
      }
      if (node.op === '-') {
        // d/dx(+/-f(x)) = +/-f'(x)
        if (node.isUnary()) {
          return new OperatorNode(node.op, node.fn, [_derivative(node.args[0], constNodes)]);
        }

        // Linearity of differentiation, d/dx(f(x) +/- g(x)) = f'(x) +/- g'(x)
        if (node.isBinary()) {
          return new OperatorNode(node.op, node.fn, [_derivative(node.args[0], constNodes), _derivative(node.args[1], constNodes)]);
        }
      }
      if (node.op === '*') {
        // d/dx(c*f(x)) = c*f'(x)
        var constantTerms = node.args.filter(function (arg) {
          return constNodes[arg] !== undefined;
        });
        if (constantTerms.length > 0) {
          var nonConstantTerms = node.args.filter(function (arg) {
            return constNodes[arg] === undefined;
          });
          var nonConstantNode = nonConstantTerms.length === 1 ? nonConstantTerms[0] : new OperatorNode('*', 'multiply', nonConstantTerms);
          var newArgs = constantTerms.concat(_derivative(nonConstantNode, constNodes));
          return new OperatorNode('*', 'multiply', newArgs);
        }

        // Product Rule, d/dx(f(x)*g(x)) = f'(x)*g(x) + f(x)*g'(x)
        return new OperatorNode('+', 'add', node.args.map(function (argOuter) {
          return new OperatorNode('*', 'multiply', node.args.map(function (argInner) {
            return argInner === argOuter ? _derivative(argInner, constNodes) : argInner.clone();
          }));
        }));
      }
      if (node.op === '/' && node.isBinary()) {
        var arg0 = node.args[0];
        var arg1 = node.args[1];

        // d/dx(f(x) / c) = f'(x) / c
        if (constNodes[arg1] !== undefined) {
          return new OperatorNode('/', 'divide', [_derivative(arg0, constNodes), arg1]);
        }

        // Reciprocal Rule, d/dx(c / f(x)) = -c(f'(x)/f(x)^2)
        if (constNodes[arg0] !== undefined) {
          return new OperatorNode('*', 'multiply', [new OperatorNode('-', 'unaryMinus', [arg0]), new OperatorNode('/', 'divide', [_derivative(arg1, constNodes), new OperatorNode('^', 'pow', [arg1.clone(), createConstantNode(2)])])]);
        }

        // Quotient rule, d/dx(f(x) / g(x)) = (f'(x)g(x) - f(x)g'(x)) / g(x)^2
        return new OperatorNode('/', 'divide', [new OperatorNode('-', 'subtract', [new OperatorNode('*', 'multiply', [_derivative(arg0, constNodes), arg1.clone()]), new OperatorNode('*', 'multiply', [arg0.clone(), _derivative(arg1, constNodes)])]), new OperatorNode('^', 'pow', [arg1.clone(), createConstantNode(2)])]);
      }
      if (node.op === '^' && node.isBinary()) {
        var _arg = node.args[0];
        var _arg2 = node.args[1];
        if (constNodes[_arg] !== undefined) {
          // If is secretly constant; 0^f(x) = 1 (in JS), 1^f(x) = 1
          if (isConstantNode(_arg) && (isZero(_arg.value) || equal(_arg.value, 1))) {
            return createConstantNode(0);
          }

          // d/dx(c^f(x)) = c^f(x)*ln(c)*f'(x)
          return new OperatorNode('*', 'multiply', [node, new OperatorNode('*', 'multiply', [new FunctionNode('log', [_arg.clone()]), _derivative(_arg2.clone(), constNodes)])]);
        }
        if (constNodes[_arg2] !== undefined) {
          if (isConstantNode(_arg2)) {
            // If is secretly constant; f(x)^0 = 1 -> d/dx(1) = 0
            if (isZero(_arg2.value)) {
              return createConstantNode(0);
            }
            // Ignore exponent; f(x)^1 = f(x)
            if (equal(_arg2.value, 1)) {
              return _derivative(_arg, constNodes);
            }
          }

          // Elementary Power Rule, d/dx(f(x)^c) = c*f'(x)*f(x)^(c-1)
          var powMinusOne = new OperatorNode('^', 'pow', [_arg.clone(), new OperatorNode('-', 'subtract', [_arg2, createConstantNode(1)])]);
          return new OperatorNode('*', 'multiply', [_arg2.clone(), new OperatorNode('*', 'multiply', [_derivative(_arg, constNodes), powMinusOne])]);
        }

        // Functional Power Rule, d/dx(f^g) = f^g*[f'*(g/f) + g'ln(f)]
        return new OperatorNode('*', 'multiply', [new OperatorNode('^', 'pow', [_arg.clone(), _arg2.clone()]), new OperatorNode('+', 'add', [new OperatorNode('*', 'multiply', [_derivative(_arg, constNodes), new OperatorNode('/', 'divide', [_arg2.clone(), _arg.clone()])]), new OperatorNode('*', 'multiply', [_derivative(_arg2, constNodes), new FunctionNode('log', [_arg.clone()])])])]);
      }
      throw new Error('Operator "' + node.op + '" is not supported by derivative, or a wrong number of arguments is passed');
    }
  });

  /**
   * Ensures the number of arguments for a function are correct,
   * and will throw an error otherwise.
   *
   * @param {FunctionNode} node
   */
  function funcArgsCheck(node) {
    // TODO add min, max etc
    if ((node.name === 'log' || node.name === 'nthRoot' || node.name === 'pow') && node.args.length === 2) {
      return;
    }

    // There should be an incorrect number of arguments if we reach here

    // Change all args to constants to avoid unidentified
    // symbol error when compiling function
    for (var i = 0; i < node.args.length; ++i) {
      node.args[i] = createConstantNode(0);
    }
    node.compile().evaluate();
    throw new Error('Expected TypeError, but none found');
  }

  /**
   * Helper function to create a constant node with a specific type
   * (number, BigNumber, Fraction)
   * @param {number} value
   * @param {string} [valueType]
   * @return {ConstantNode}
   */
  function createConstantNode(value, valueType) {
    return new ConstantNode(numeric(value, valueType || config.number));
  }
  return derivative;
});