mathjs/src/function/algebra/simplify.js

import { isConstantNode, isParenthesisNode } from '../../utils/is'
import { factory } from '../../utils/factory'
import { createUtil } from './simplify/util'
import { createSimplifyCore } from './simplify/simplifyCore'
import { createSimplifyConstant } from './simplify/simplifyConstant'
import { createResolve } from './simplify/resolve'
import { hasOwnProperty } from '../../utils/object'

const name = 'simplify'
const dependencies = [
  'config',
  'typed',
  'parse',
  'add',
  'subtract',
  'multiply',
  'divide',
  'pow',
  'isZero',
  'equal',
  '?fraction',
  '?bignumber',
  'mathWithTransform',
  'ConstantNode',
  'FunctionNode',
  'OperatorNode',
  'ParenthesisNode',
  'SymbolNode'
]

export const createSimplify = /* #__PURE__ */ factory(name, dependencies, (
  {
    config,
    typed,
    parse,
    add,
    subtract,
    multiply,
    divide,
    pow,
    isZero,
    equal,
    fraction,
    bignumber,
    mathWithTransform,
    ConstantNode,
    FunctionNode,
    OperatorNode,
    ParenthesisNode,
    SymbolNode
  }
) => {
  const simplifyConstant = createSimplifyConstant({
    typed,
    config,
    mathWithTransform,
    fraction,
    bignumber,
    ConstantNode,
    OperatorNode,
    FunctionNode,
    SymbolNode
  })
  const simplifyCore = createSimplifyCore({
    equal,
    isZero,
    add,
    subtract,
    multiply,
    divide,
    pow,
    ConstantNode,
    OperatorNode,
    FunctionNode,
    ParenthesisNode
  })
  const resolve = createResolve({
    parse,
    FunctionNode,
    OperatorNode,
    ParenthesisNode
  })

  const { isCommutative, isAssociative, flatten, unflattenr, unflattenl, createMakeNodeFunction } =
    createUtil({ FunctionNode, OperatorNode, SymbolNode })

  /**
   * Simplify an expression tree.
   *
   * A list of rules are applied to an expression, repeating over the list until
   * no further changes are made.
   * It's possible to pass a custom set of rules to the function as second
   * argument. A rule can be specified as an object, string, or function:
   *
   *     const rules = [
   *       { l: 'n1*n3 + n2*n3', r: '(n1+n2)*n3' },
   *       'n1*n3 + n2*n3 -> (n1+n2)*n3',
   *       function (node) {
   *         // ... return a new node or return the node unchanged
   *         return node
   *       }
   *     ]
   *
   * String and object rules consist of a left and right pattern. The left is
   * used to match against the expression and the right determines what matches
   * are replaced with. The main difference between a pattern and a normal
   * expression is that variables starting with the following characters are
   * interpreted as wildcards:
   *
   * - 'n' - matches any Node
   * - 'c' - matches any ConstantNode
   * - 'v' - matches any Node that is not a ConstantNode
   *
   * The default list of rules is exposed on the function as `simplify.rules`
   * and can be used as a basis to built a set of custom rules.
   *
   * For more details on the theory, see:
   *
   * - [Strategies for simplifying math expressions (Stackoverflow)](https://stackoverflow.com/questions/7540227/strategies-for-simplifying-math-expressions)
   * - [Symbolic computation - Simplification (Wikipedia)](https://en.wikipedia.org/wiki/Symbolic_computation#Simplification)
   *
   *  An optional `options` argument can be passed as last argument of `simplify`.
   *  There is currently one option available: `exactFractions`, a boolean which
   *  is `true` by default.
   *
   * Syntax:
   *
   *     simplify(expr)
   *     simplify(expr, rules)
   *     simplify(expr, rules)
   *     simplify(expr, rules, scope)
   *     simplify(expr, rules, scope, options)
   *     simplify(expr, scope)
   *     simplify(expr, scope, options)
   *
   * Examples:
   *
   *     math.simplify('2 * 1 * x ^ (2 - 1)')      // Node "2 * x"
   *     math.simplify('2 * 3 * x', {x: 4})        // Node "24"
   *     const f = math.parse('2 * 1 * x ^ (2 - 1)')
   *     math.simplify(f)                          // Node "2 * x"
   *     math.simplify('0.4 * x', {}, {exactFractions: true})  // Node "x * 2 / 5"
   *     math.simplify('0.4 * x', {}, {exactFractions: false}) // Node "0.4 * x"
   *
   * See also:
   *
   *     derivative, parse, evaluate, rationalize
   *
   * @param {Node | string} expr
   *            The expression to be simplified
   * @param {Array<{l:string, r: string} | string | function>} [rules]
   *            Optional list with custom rules
   * @return {Node} Returns the simplified form of `expr`
   */
  const simplify = typed('simplify', {
    string: function (expr) {
      return simplify(parse(expr), simplify.rules, {}, {})
    },

    'string, Object': function (expr, scope) {
      return simplify(parse(expr), simplify.rules, scope, {})
    },

    'string, Object, Object': function (expr, scope, options) {
      return simplify(parse(expr), simplify.rules, scope, options)
    },

    'string, Array': function (expr, rules) {
      return simplify(parse(expr), rules, {}, {})
    },

    'string, Array, Object': function (expr, rules, scope) {
      return simplify(parse(expr), rules, scope, {})
    },

    'string, Array, Object, Object': function (expr, rules, scope, options) {
      return simplify(parse(expr), rules, scope, options)
    },

    'Node, Object': function (expr, scope) {
      return simplify(expr, simplify.rules, scope, {})
    },

    'Node, Object, Object': function (expr, scope, options) {
      return simplify(expr, simplify.rules, scope, options)
    },

    Node: function (expr) {
      return simplify(expr, simplify.rules, {}, {})
    },

    'Node, Array': function (expr, rules) {
      return simplify(expr, rules, {}, {})
    },

    'Node, Array, Object': function (expr, rules, scope) {
      return simplify(expr, rules, scope, {})
    },

    'Node, Array, Object, Object': function (expr, rules, scope, options) {
      rules = _buildRules(rules)
      let res = resolve(expr, scope)
      res = removeParens(res)
      const visited = {}
      let str = res.toString({ parenthesis: 'all' })
      while (!visited[str]) {
        visited[str] = true
        _lastsym = 0 // counter for placeholder symbols
        for (let i = 0; i < rules.length; i++) {
          if (typeof rules[i] === 'function') {
            res = rules[i](res, options)
          } else {
            flatten(res)
            res = applyRule(res, rules[i])
          }
          unflattenl(res) // using left-heavy binary tree here since custom rule functions may expect it
        }
        str = res.toString({ parenthesis: 'all' })
      }
      return res
    }
  })
  simplify.simplifyCore = simplifyCore
  simplify.resolve = resolve

  function removeParens (node) {
    return node.transform(function (node, path, parent) {
      return isParenthesisNode(node)
        ? removeParens(node.content)
        : node
    })
  }

  // All constants that are allowed in rules
  const SUPPORTED_CONSTANTS = {
    true: true,
    false: true,
    e: true,
    i: true,
    Infinity: true,
    LN2: true,
    LN10: true,
    LOG2E: true,
    LOG10E: true,
    NaN: true,
    phi: true,
    pi: true,
    SQRT1_2: true,
    SQRT2: true,
    tau: true
    // null: false,
    // undefined: false,
    // version: false,
  }

  // Array of strings, used to build the ruleSet.
  // Each l (left side) and r (right side) are parsed by
  // the expression parser into a node tree.
  // Left hand sides are matched to subtrees within the
  // expression to be parsed and replaced with the right
  // hand side.
  // TODO: Add support for constraints on constants (either in the form of a '=' expression or a callback [callback allows things like comparing symbols alphabetically])
  // 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.
  // It is possible to get into an infinite loop with conflicting rules
  simplify.rules = [
    simplifyCore,
    // { l: 'n+0', r: 'n' },     // simplifyCore
    // { l: 'n^0', r: '1' },     // simplifyCore
    // { l: '0*n', r: '0' },     // simplifyCore
    // { l: 'n/n', r: '1'},      // simplifyCore
    // { l: 'n^1', r: 'n' },     // simplifyCore
    // { l: '+n1', r:'n1' },     // simplifyCore
    // { l: 'n--n1', r:'n+n1' }, // simplifyCore
    { l: 'log(e)', r: '1' },

    // temporary rules
    { l: 'n-n1', r: 'n+-n1' }, // temporarily replace 'subtract' so we can further flatten the 'add' operator
    { l: '-(c*v)', r: '(-c) * v' }, // make non-constant terms positive
    { l: '-v', r: '(-1) * v' },
    { l: 'n/n1^n2', r: 'n*n1^-n2' }, // temporarily replace 'divide' so we can further flatten the 'multiply' operator
    { l: 'n/n1', r: 'n*n1^-1' },

    // expand nested exponentiation
    { l: '(n ^ n1) ^ n2', r: 'n ^ (n1 * n2)' },

    // collect like factors
    { l: 'n*n', r: 'n^2' },
    { l: 'n * n^n1', r: 'n^(n1+1)' },
    { l: 'n^n1 * n^n2', r: 'n^(n1+n2)' },

    // collect like terms
    { l: 'n+n', r: '2*n' },
    { l: 'n+-n', r: '0' },
    { l: 'n1*n2 + n2', r: '(n1+1)*n2' },
    { l: 'n1*n3 + n2*n3', r: '(n1+n2)*n3' },

    // remove parenthesis in the case of negating a quantitiy
    { l: 'n1 + -1 * (n2 + n3)', r: 'n1 + -1 * n2 + -1 * n3' },

    simplifyConstant,

    { l: '(-n)*n1', r: '-(n*n1)' }, // make factors positive (and undo 'make non-constant terms positive')

    // ordering of constants
    { l: 'c+v', r: 'v+c', context: { add: { commutative: false } } },
    { l: 'v*c', r: 'c*v', context: { multiply: { commutative: false } } },

    // undo temporary rules
    // { l: '(-1) * n', r: '-n' }, // #811 added test which proved this is redundant
    { l: 'n+-n1', r: 'n-n1' }, // undo replace 'subtract'
    { l: 'n*(n1^-1)', r: 'n/n1' }, // undo replace 'divide'
    { l: 'n*n1^-n2', r: 'n/n1^n2' },
    { l: 'n1^-1', r: '1/n1' },

    { l: 'n*(n1/n2)', r: '(n*n1)/n2' }, // '*' before '/'
    { l: 'n-(n1+n2)', r: 'n-n1-n2' }, // '-' before '+'
    // { l: '(n1/n2)/n3', r: 'n1/(n2*n3)' },
    // { l: '(n*n1)/(n*n2)', r: 'n1/n2' },

    { l: '1*n', r: 'n' } // this pattern can be produced by simplifyConstant

  ]

  /**
   * Parse the string array of rules into nodes
   *
   * Example syntax for rules:
   *
   * Position constants to the left in a product:
   * { l: 'n1 * c1', r: 'c1 * n1' }
   * n1 is any Node, and c1 is a ConstantNode.
   *
   * Apply difference of squares formula:
   * { l: '(n1 - n2) * (n1 + n2)', r: 'n1^2 - n2^2' }
   * n1, n2 mean any Node.
   *
   * Short hand notation:
   * 'n1 * c1 -> c1 * n1'
   */
  function _buildRules (rules) {
    // Array of rules to be used to simplify expressions
    const ruleSet = []
    for (let i = 0; i < rules.length; i++) {
      let rule = rules[i]
      let newRule
      const ruleType = typeof rule
      switch (ruleType) {
        case 'string':
        {
          const lr = rule.split('->')
          if (lr.length === 2) {
            rule = { l: lr[0], r: lr[1] }
          } else {
            throw SyntaxError('Could not parse rule: ' + rule)
          }
        }
        /* falls through */
        case 'object':
          newRule = {
            l: removeParens(parse(rule.l)),
            r: removeParens(parse(rule.r))
          }
          if (rule.context) {
            newRule.evaluate = rule.context
          }
          if (rule.evaluate) {
            newRule.evaluate = parse(rule.evaluate)
          }

          if (isAssociative(newRule.l)) {
            const makeNode = createMakeNodeFunction(newRule.l)
            const expandsym = _getExpandPlaceholderSymbol()
            newRule.expanded = {}
            newRule.expanded.l = makeNode([newRule.l.clone(), expandsym])
            // Push the expandsym into the deepest possible branch.
            // This helps to match the newRule against nodes returned from getSplits() later on.
            flatten(newRule.expanded.l)
            unflattenr(newRule.expanded.l)
            newRule.expanded.r = makeNode([newRule.r, expandsym])
          }
          break
        case 'function':
          newRule = rule
          break
        default:
          throw TypeError('Unsupported type of rule: ' + ruleType)
      }
      // console.log('Adding rule: ' + rules[i])
      // console.log(newRule)
      ruleSet.push(newRule)
    }
    return ruleSet
  }

  let _lastsym = 0
  function _getExpandPlaceholderSymbol () {
    return new SymbolNode('_p' + _lastsym++)
  }

  /**
   * Returns a simplfied form of node, or the original node if no simplification was possible.
   *
   * @param  {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
   * @return {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} The simplified form of `expr`, or the original node if no simplification was possible.
   */
  const applyRule = typed('applyRule', {
    'Node, Object': function (node, rule) {
      // console.log('Entering applyRule(' + node.toString() + ')')

      // Do not clone node unless we find a match
      let res = node

      // First replace our child nodes with their simplified versions
      // If a child could not be simplified, the assignments will have
      // no effect since the node is returned unchanged
      if (res instanceof OperatorNode || res instanceof FunctionNode) {
        if (res.args) {
          for (let i = 0; i < res.args.length; i++) {
            res.args[i] = applyRule(res.args[i], rule)
          }
        }
      } else if (res instanceof ParenthesisNode) {
        if (res.content) {
          res.content = applyRule(res.content, rule)
        }
      }

      // Try to match a rule against this node
      let repl = rule.r
      let matches = _ruleMatch(rule.l, res)[0]

      // If the rule is associative operator, we can try matching it while allowing additional terms.
      // 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.
      if (!matches && rule.expanded) {
        repl = rule.expanded.r
        matches = _ruleMatch(rule.expanded.l, res)[0]
      }

      if (matches) {
        // const before = res.toString({parenthesis: 'all'})

        // Create a new node by cloning the rhs of the matched rule
        // we keep any implicit multiplication state if relevant
        const implicit = res.implicit
        res = repl.clone()
        if (implicit && 'implicit' in repl) {
          res.implicit = true
        }

        // Replace placeholders with their respective nodes without traversing deeper into the replaced nodes
        res = res.transform(function (node) {
          if (node.isSymbolNode && hasOwnProperty(matches.placeholders, node.name)) {
            return matches.placeholders[node.name].clone()
          } else {
            return node
          }
        })

        // const after = res.toString({parenthesis: 'all'})
        // console.log('Simplified ' + before + ' to ' + after)
      }

      return res
    }
  })

  /**
   * Get (binary) combinations of a flattened binary node
   * e.g. +(node1, node2, node3) -> [
   *        +(node1,  +(node2, node3)),
   *        +(node2,  +(node1, node3)),
   *        +(node3,  +(node1, node2))]
   *
   */
  function getSplits (node, context) {
    const res = []
    let right, rightArgs
    const makeNode = createMakeNodeFunction(node)
    if (isCommutative(node, context)) {
      for (let i = 0; i < node.args.length; i++) {
        rightArgs = node.args.slice(0)
        rightArgs.splice(i, 1)
        right = (rightArgs.length === 1) ? rightArgs[0] : makeNode(rightArgs)
        res.push(makeNode([node.args[i], right]))
      }
    } else {
      rightArgs = node.args.slice(1)
      right = (rightArgs.length === 1) ? rightArgs[0] : makeNode(rightArgs)
      res.push(makeNode([node.args[0], right]))
    }
    return res
  }

  /**
   * Returns the set union of two match-placeholders or null if there is a conflict.
   */
  function mergeMatch (match1, match2) {
    const res = { placeholders: {} }

    // Some matches may not have placeholders; this is OK
    if (!match1.placeholders && !match2.placeholders) {
      return res
    } else if (!match1.placeholders) {
      return match2
    } else if (!match2.placeholders) {
      return match1
    }

    // Placeholders with the same key must match exactly
    for (const key in match1.placeholders) {
      res.placeholders[key] = match1.placeholders[key]
      if (hasOwnProperty(match2.placeholders, key)) {
        if (!_exactMatch(match1.placeholders[key], match2.placeholders[key])) {
          return null
        }
      }
    }

    for (const key in match2.placeholders) {
      res.placeholders[key] = match2.placeholders[key]
    }

    return res
  }

  /**
   * Combine two lists of matches by applying mergeMatch to the cartesian product of two lists of matches.
   * Each list represents matches found in one child of a node.
   */
  function combineChildMatches (list1, list2) {
    const res = []

    if (list1.length === 0 || list2.length === 0) {
      return res
    }

    let merged
    for (let i1 = 0; i1 < list1.length; i1++) {
      for (let i2 = 0; i2 < list2.length; i2++) {
        merged = mergeMatch(list1[i1], list2[i2])
        if (merged) {
          res.push(merged)
        }
      }
    }
    return res
  }

  /**
   * Combine multiple lists of matches by applying mergeMatch to the cartesian product of two lists of matches.
   * Each list represents matches found in one child of a node.
   * Returns a list of unique matches.
   */
  function mergeChildMatches (childMatches) {
    if (childMatches.length === 0) {
      return childMatches
    }

    const sets = childMatches.reduce(combineChildMatches)
    const uniqueSets = []
    const unique = {}
    for (let i = 0; i < sets.length; i++) {
      const s = JSON.stringify(sets[i])
      if (!unique[s]) {
        unique[s] = true
        uniqueSets.push(sets[i])
      }
    }
    return uniqueSets
  }

  /**
   * Determines whether node matches rule.
   *
   * @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() + ')')
    let 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
        const childMatches = []
        for (let i = 0; i < rule.args.length; i++) {
          const 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.
        const splits = getSplits(node, rule.context)
        let splitMatches = []
        for (let i = 0; i < splits.length; i++) {
          const 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 (SUPPORTED_CONSTANTS[rule.name]) {
        // 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 (!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 (!equal(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 (!equal(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 (let i = 0; i < p.args.length; i++) {
        if (!_exactMatch(p.args[i], q.args[i])) {
          return false
        }
      }
    } else {
      return false
    }

    return true
  }

  return simplify
})