docs:Better basic examples.

Author: antononcube

README-work.md

@@ -124,43 +124,67 @@ flowchart TD
 
 Make a new computation engine object and evaluate a LaTeX expression:
 
-```raku
+```raku, results=asis
 use CortexJS;
-my $ce = ComputeEngine.new;
 
-$ce.evaluate($ce.parse-latex('e^{i\\pi}'))
+'e^{i\\pi}' ==> evaluate()
 ```
 
 Expand expression:
 
-```raku
-$ce.to-latex($ce.expand($ce.parse-latex('(a + b)^2')));
+```raku, results=asis
+'(a + b)^2' ==> expand()
 ```
 
 Simplify expression:
 
+```raku, results=asis
+my $expr = parse-latex('3x^2 + 2x^2 + x + 5');
+say "{to-latex($expr)} = {to-latex(simplify($expr))}";
+```
+
+Solve a symbolic equation
+
+```raku, results=asis
+'x^2 - a x + 1 = 0' ==> solve(vars => 'a')
+```
+
+The input can be both a LaTeX string (as above) or in [MathJSON format](https://mathlive.io/math-json/). 
+Here the LaTeX expression `(a - b)^2` is converted to MathJSON using the sub `parse-latex`:
+
 ```raku
-my $expr = $ce.parse-latex('3x^2 + 2x^2 + x + 5');
-say "{$ce.to-latex($expr)} = {$ce.to-latex($ce.simplify($expr))}";
+parse-latex('(a - b)^2').raku 
 ```
 
-Using assignment for repeated expression evaluation:
+Here the corresponding MathJSON expression is expanded:
 
 ```raku
-my $expr = $ce.parse-latex("3x^2+4x+2");
+["Power", ["Add", "a", ["Negate", "b"]], 2]
+==> expand()
+```
 
-for (0, 0.1 ... 1) -> $x {
-  $ce.assign('x', $x);
-  say "f($x) = {$ce.evaluate($expr)}";
-}
+MathJSON expressions can be converted to LaTeX with `to-latex`:
+
+```raku, results=asis
+_ ==> to-latex()
 ```
 
-Can be put in the last code block:
+**Remark** The "free functions" `evaluate`, `N`, `simplify`, `assign`, `expand`, `expandAll`, `factor`, and `solve`
+try to recognize (or parse) a string input as LaTeX code, and if parsing is successful, 
+then (more or less) the processing pipeline in applied: `$expr ==> parse-latex() ==> &func() ==> to-latex()`.
+
+Using assignment for repeated expression evaluation:
 
 ```raku
-LEAVE $ce.close;
+my $expr = parse-latex("3x^2+4x+2");
+
+for (0, 0.1 ... 1) -> $x {
+  assign('x', $x);
+  say "f($x) = {evaluate($expr)}";
+}
 ```
 
+
 ----
 
 ## References

README.md

@@ -124,67 +124,95 @@ flowchart TD
 
 Make a new computation engine object and evaluate a LaTeX expression:
 
-```raku
+```raku, results=asis
 use CortexJS;
-my $ce = ComputeEngine.new;
 
-$ce.evaluate($ce.parse-latex('e^{i\\pi}'))
+'e^{i\\pi}' ==> evaluate()
 ```
+$-1$
+
+
+Expand expression:
+
+```raku, results=asis
+'(a + b)^2' ==> expand()
 ```
-# -1
+$a^2+b^2+2ab$
+
+
+Simplify expression:
+
+```raku, results=asis
+my $expr = parse-latex('3x^2 + 2x^2 + x + 5');
+say "{to-latex($expr)} = {to-latex(simplify($expr))}";
 ```
+$2x^2+3x^2+x+5$ = $5x^2+x+5$
 
-Expand expression:
+
+Solve a symbolic equation
+
+```raku, results=asis
+'x^2 - a x + 1 = 0' ==> solve(vars => 'a')
+```
+$x+\frac{1}{x}$
+
+
+The input can be both a LaTeX string (as above) or in [MathJSON format](https://mathlive.io/math-json/). 
+Here the LaTeX expression `(a - b)^2` is converted to MathJSON using the sub `parse-latex`:
 
 ```raku
-$ce.to-latex($ce.expand($ce.parse-latex('(a + b)^2')));
+parse-latex('(a - b)^2').raku 
 ```
 ```
-# a^2+b^2+2ab
+# $["Power", ["Add", "a", ["Negate", "b"]], 2]
 ```
 
-Simplify expression:
+Here the corresponding MathJSON expression is expanded:
 
 ```raku
-my $expr = $ce.parse-latex('3x^2 + 2x^2 + x + 5');
-say "{$ce.to-latex($expr)} = {$ce.to-latex($ce.simplify($expr))}";
+["Power", ["Add", "a", ["Negate", "b"]], 2]
+==> expand()
 ```
 ```
-# 2x^2+3x^2+x+5 = 5x^2+x+5
+# [Add [Power a 2] [Power b 2] [Multiply -2 a b]]
 ```
 
+MathJSON expressions can be converted to LaTeX with `to-latex`:
+
+```raku, results=asis
+_ ==> to-latex()
+```
+$a^2+b^2-2ab$
+
+
+**Remark** The "free functions" `evaluate`, `N`, `simplify`, `assign`, `expand`, `expandAll`, `factor`, and `solve`
+try to recognize (or parse) a string input as LaTeX code, and if parsing is successful, 
+then (more or less) the processing pipeline in applied: `$expr ==> parse-latex() ==> &func() ==> to-latex()`.
+
 Using assignment for repeated expression evaluation:
 
 ```raku
-my $expr = $ce.parse-latex("3x^2+4x+2");
+my $expr = parse-latex("3x^2+4x+2");
 
 for (0, 0.1 ... 1) -> $x {
-  $ce.assign('x', $x);
-  say "f($x) = {$ce.evaluate($expr)}";
+  assign('x', $x);
+  say "f($x) = {evaluate($expr)}";
 }
 ```
 ```
-# f(0) = 2
-# f(0.1) = 2.43
-# f(0.2) = 2.92
-# f(0.3) = 3.47
-# f(0.4) = 4.08
-# f(0.5) = 4.75
-# f(0.6) = 5.48
-# f(0.7) = 6.27
-# f(0.8) = 7.12
-# f(0.9) = 8.03
-# f(1) = 9
+# f(0) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.1) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.2) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.3) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.4) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.5) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.6) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.7) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.8) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(0.9) = Add Multiply 3 Power x 2 Multiply 4 x 2
+# f(1) = Add Multiply 3 Power x 2 Multiply 4 x 2
 ```
 
-Can be put in the last code block:
-
-```raku
-LEAVE $ce.close;
-```
-```
-# (Any)
-```
 
 ----