WIP: "symbolic" evaluation

examples/symbolic_evaluation.js

@@ -0,0 +1,77 @@
+const math = require('..')
+function symval (expr, scope = {}) {
+  return math.simplify(
+    expr, [
+      'n1-n2 -> n1+(-n2)',
+      'n1/n2 -> n1*n2^(-1)',
+      math.simplify.simplifyConstant,
+      'n1*n2^(-1) -> n1/n2',
+      'n1+(-n2) -> n1-n2'
+    ],
+    scope, {
+      unwrapConstants: true
+    }
+  )
+}
+
+function mystringify (obj) {
+  let s = '{'
+  for (const key in obj) {
+    s += `${key}: ${obj[key]}, `
+  }
+  return s.slice(0, -2) + '}'
+}
+
+function logExample (expr, scope = {}) {
+  let header = `Evaluating: '${expr}'`
+  if (Object.keys(scope).length > 0) {
+    header += ` in scope ${mystringify(scope)}`
+  }
+  console.log(header)
+  let result = symval(expr, scope)
+  if (math.isNode(result)) {
+    result = `Expression ${result.toString()}`
+  }
+  console.log(`  --> ${result}`)
+}
+
+let point = 1
+console.log(`${point++}. By just simplifying constants as fully as possible, using
+the scope as necessary, we create a sort of "symbolic" evaluation:`)
+logExample('x*y + 3x - y + 2', { y: 7 })
+console.log(`
+${point++}. If all of the free variables have values, this evaluates
+all the way to the numeric value:`)
+logExample('x*y + 3x - y + 2', { x: 1, y: 7 })
+console.log(`
+${point++}. It works with matrices as well, for example`)
+logExample('[x^2 + 3x + x*y, y, 12]', { x: 2 })
+logExample('[x^2 + 3x + x*y, y, 12]', { x: 2, y: 7 })
+console.log(`(Note all the fractions because currently simplifyConstant prefers
+them. That preference could be tweaked for this purpose.)
+
+${point++}. This lets you more easily perform operations like symbolic differentiation:`)
+logExample('derivative(sin(x) + exp(x) + x^3, x)')
+console.log("(Note no quotes in the argument to 'derivative' -- it is directly\n" +
+  'operating on the expression, without any string values involved.)')
+
+console.log(`
+${point++}. You can also build up expressions incrementally:`)
+logExample('derivative(h3,x)', {
+  h3: symval('h1+h2'),
+  h1: symval('x^2+3x'),
+  h2: symval('3x+7')
+})
+console.log(`
+${point++}. Some kinks still remain at the moment. The scope is not affected
+by assignment expressions, and scope values for the variable of differentiation
+disrupt the results:`)
+logExample('derivative(x^3 + x^2, x)')
+logExample('derivative(x^3 + x^2, x)', { x: 1 })
+console.log(`${''}(We'd like the latter evaluation to return the result of the
+first differentiation, evaluated at 1, or namely 5. However, there is not (yet)
+a concept in mathjs (specifically in 'resolve') that  'derivative' creates a
+variable-binding environment, blocking off the 'x' from being substituted via
+the outside scope within its first argument.)
+
+But such features can be implemented.`)

src/function/algebra/simplify.js

@@ -1,4 +1,4 @@
-import { isConstantNode, isParenthesisNode } from '../../utils/is.js'
+import { isNode, isConstantNode, isParenthesisNode } from '../../utils/is.js'
 import { factory } from '../../utils/factory.js'
 import { createUtil } from './simplify/util.js'
 import { createSimplifyConstant } from './simplify/simplifyConstant.js'
@@ -156,6 +156,8 @@ export const createSimplify = /* #__PURE__ */ factory(name, dependencies, (
    *  - `fractionsLimit` (10000): when `exactFractions` is true, constants will
    *    be expressed as fractions only when both numerator and denominator
    *    are smaller than `fractionsLimit`.
+   *  - `unwrapConstants` (false): if the entire expression simplifies down to
+   *    a constant, return the value directly (as opposed to wrapped in a Node).
    *
    * Syntax:
    *
@@ -266,6 +268,7 @@ export const createSimplify = /* #__PURE__ */ factory(name, dependencies, (
               laststr = newstr
             }
           }
+          if (!isNode(res)) return res // short-circuit if we got to a concrete value
           /* Use left-heavy binary tree internally,
            * since custom rule functions may expect it
            */
@@ -279,6 +282,7 @@ export const createSimplify = /* #__PURE__ */ factory(name, dependencies, (
   simplify.defaultContext = defaultContext
   simplify.realContext = realContext
   simplify.positiveContext = positiveContext
+  simplify.simplifyConstant = simplifyConstant
 
   function removeParens (node) {
     return node.transform(function (node, path, parent) {

src/function/algebra/simplify/simplifyConstant.js

@@ -42,7 +42,9 @@ export const createSimplifyConstant = /* #__PURE__ */ factory(name, dependencies
     createUtil({ FunctionNode, OperatorNode, SymbolNode })
 
   function simplifyConstant (expr, options) {
-    return _ensureNode(foldFraction(expr, options))
+    const folded = foldFraction(expr, options)
+    if (options.unwrapConstants) return folded
+    return _ensureNode(folded)
   }
 
   function _removeFractions (thing) {
@@ -310,12 +312,10 @@ export const createSimplifyConstant = /* #__PURE__ */ factory(name, dependencies
           if (operatorFunctions.indexOf(node.name) === -1) {
             const args = node.args.map(arg => foldFraction(arg, options))
 
-            // If all args are numbers
-            if (!args.some(isNode)) {
-              try {
-                return _eval(node.name, args, options)
-              } catch (ignoreandcontinue) { }
-            }
+            // If the function can handle the arguments, call it
+            try {
+              return _eval(node.name, args, options)
+            } catch (ignoreandcontinue) { }
 
             // Size of a matrix does not depend on entries
             if (node.name === 'size' &&