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.  And `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 `a s' is 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 `a s'.  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'.