10.11.11 Automatic Rewrites

It is possible to get Calc to apply a set of rewrite rules on all results, effectively adding to the built-in set of default simplifications. To do this, simply store your rule set in the variable EvalRules. There is a convenient s E command for editing EvalRules; see Operations on Variables.

For example, suppose you want ‘sin(a + b)’ to be expanded out to ‘sin(b) cos(a) + cos(b) sin(a)’ wherever it appears, and similarly for ‘cos(a + b)’. The corresponding rewrite rule set would be,

[ sin(a + b)  :=  cos(a) sin(b) + sin(a) cos(b),
  cos(a + b)  :=  cos(a) cos(b) - sin(a) sin(b) ]

To apply these manually, you could put them in a variable called trigexp and then use a r trigexp every time you wanted to expand trig functions. But if instead you store them in the variable EvalRules, they will automatically be applied to all sines and cosines of sums. Then, with ‘2 x’ and ‘45’ on the stack, typing + S will (assuming Degrees mode) result in ‘0.7071 sin(2 x) + 0.7071 cos(2 x)’ automatically.

As each level of a formula is evaluated, the rules from EvalRules are applied before the default simplifications. Rewriting continues until no further EvalRules apply. Note that this is different from the usual order of application of rewrite rules: EvalRules works from the bottom up, simplifying the arguments to a function before the function itself, while a r applies rules from the top down.

Because the EvalRules are tried first, you can use them to override the normal behavior of any built-in Calc function.

It is important not to write a rule that will get into an infinite loop. For example, the rule set ‘[f(0) := 1, f(n) := n f(n-1)]’ appears to be a good definition of a factorial function, but it is unsafe. Imagine what happens if ‘f(2.5)’ is simplified. Calc will continue to subtract 1 from this argument forever without reaching zero. A safer second rule would be ‘f(n) := n f(n-1) :: n>0’. Another dangerous rule is ‘g(x, y) := g(y, x)’. Rewriting ‘g(2, 4)’, this would bounce back and forth between that and ‘g(4, 2)’ forever. If an infinite loop in EvalRules occurs, Emacs will eventually stop with a “Computation got stuck or ran too long” message.

Another subtle difference between EvalRules and regular rewrites concerns rules that rewrite a formula into an identical formula. For example, ‘f(n) := f(floor(n))’ “fails to match” when ‘n’ is already an integer. But in EvalRules this case is detected only if the righthand side literally becomes the original formula before any further simplification. This means that ‘f(n) := f(floor(n))’ will get into an infinite loop if it occurs in EvalRules. Calc will replace ‘f(6)’ with ‘f(floor(6))’, which is different from ‘f(6)’, so it will consider the rule to have matched and will continue simplifying that formula; first the argument is simplified to get ‘f(6)’, then the rule matches again to get ‘f(floor(6))’ again, ad infinitum. A much safer rule would check its argument first, say, with ‘f(n) := f(floor(n)) :: !dint(n)’.

(What really happens is that the rewrite mechanism substitutes the meta-variables in the righthand side of a rule, compares to see if the result is the same as the original formula and fails if so, then uses the default simplifications to simplify the result and compares again (and again fails if the formula has simplified back to its original form). The only special wrinkle for the EvalRules is that the same rules will come back into play when the default simplifications are used. What Calc wants to do is build ‘f(floor(6))’, see that this is different from the original formula, simplify to ‘f(6)’, see that this is the same as the original formula, and thus halt the rewriting. But while simplifying, ‘f(6)’ will again trigger the same EvalRules rule and Calc will get into a loop inside the rewrite mechanism itself.)

The phase, schedule, and iterations markers do not work in EvalRules. If the rule set is divided into phases, only the phase 1 rules are applied, and the schedule is ignored. The rules are always repeated as many times as possible.

The EvalRules are applied to all function calls in a formula, but not to numbers (and other number-like objects like error forms), nor to vectors or individual variable names. (Though they will apply to components of vectors and error forms when appropriate.) You might try to make a variable phihat which automatically expands to its definition without the need to press = by writing the rule ‘quote(phihat) := (1-sqrt(5))/2’, but unfortunately this rule will not work as part of EvalRules.

Finally, another limitation is that Calc sometimes calls its built-in functions directly rather than going through the default simplifications. When it does this, EvalRules will not be able to override those functions. For example, when you take the absolute value of the complex number ‘(2, 3)’, Calc computes ‘sqrt(2*2 + 3*3)’ by calling the multiplication, addition, and square root functions directly rather than applying the default simplifications to this formula. So an EvalRules rule that (perversely) rewrites ‘sqrt(13) := 6’ would not apply. (However, if you put Calc into Symbolic mode so that ‘sqrt(13)’ will be left in symbolic form by the built-in square root function, your rule will be able to apply. But if the complex number were ‘(3,4)’, so that ‘sqrt(25)’ must be calculated, then Symbolic mode will not help because ‘sqrt(25)’ can be evaluated exactly to 5.)

One subtle restriction that normally only manifests itself with EvalRules is that while a given rewrite rule is in the process of being checked, that same rule cannot be recursively applied. Calc effectively removes the rule from its rule set while checking the rule, then puts it back once the match succeeds or fails. (The technical reason for this is that compiled pattern programs are not reentrant.) For example, consider the rule ‘foo(x) := x :: foo(x/2) > 0’ attempting to match ‘foo(8)’. This rule will be inactive while the condition ‘foo(4) > 0’ is checked, even though it might be an integral part of evaluating that condition. Note that this is not a problem for the more usual recursive type of rule, such as ‘foo(x) := foo(x/2)’, because there the rule has succeeded and been reactivated by the time the righthand side is evaluated.

If EvalRules has no stored value (its default state), or if anything but a vector is stored in it, then it is ignored.

Even though Calc’s rewrite mechanism is designed to compare rewrite rules to formulas as quickly as possible, storing rules in EvalRules may make Calc run substantially slower. This is particularly true of rules where the top-level call is a commonly used function, or is not fixed. The rule ‘f(n) := n f(n-1) :: n>0’ will only activate the rewrite mechanism for calls to the function f, but ‘lg(n) + lg(m) := lg(n m)’ will check every ‘+’ operator.

apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])

may seem more “efficient” than two separate rules for ln and log10, but actually it is vastly less efficient because rules with apply as the top-level pattern must be tested against every function call that is simplified.

Suppose you want ‘sin(a + b)’ to be expanded out not all the time, but only when algebraic simplifications are used to simplify the formula. The variable AlgSimpRules holds rules for this purpose. The a s command will apply EvalRules and AlgSimpRules to the formula, as well as all of its built-in simplifications.

Most of the special limitations for EvalRules don’t apply to AlgSimpRules. Calc simply does an a r AlgSimpRules command with an infinite repeat count as the first step of algebraic simplifications. It then applies its own built-in simplifications throughout the formula, and then repeats these two steps (along with applying the default simplifications) until no further changes are possible.

There are also ExtSimpRules and UnitSimpRules variables that are used by a e and u s, respectively; these commands also apply EvalRules and AlgSimpRules. The variable IntegSimpRules contains simplification rules that are used only during integration by a i.