"Unsafe" Simplifications
------------------------

The `a e' (`calc-simplify-extended') [`esimplify'] command is like `a
s' except that it applies some additional simplifications which are
not "safe" in all cases.  Use this only if you know the values in your
formula lie in the restricted ranges for which these simplifications
are valid.  The symbolic integrator uses `a e'; one effect of this is
that the integrator's results must be used with caution.  Where an
integral table will often attach conditions like "for positive `a'
only," Calc (like most other symbolic integration programs) will
simply produce an unqualified result.

Because `a e''s simplifications are unsafe, it is sometimes better to
type `C-u -3 a v', which does extended simplification only on the top
level of the formula without affecting the sub-formulas.  In fact,
`C-u -3 j v' allows you to target extended simplification to any
specific part of a formula.

The variable `ExtSimpRules' contains rewrites to be applied by the `a
e' command.  These are applied in addition to `EvalRules' and
`AlgSimpRules'.  (The `a r AlgSimpRules' step described above is
simply followed by an `a r ExtSimpRules' step.)

Following is a complete list of "unsafe" simplifications performed by
`a e'.


Inverse trigonometric or hyperbolic functions, called with their
corresponding non-inverse functions as arguments, are simplified by `a
e'.  For example, `arcsin(sin(x))' changes to `x'.  Also,
`arcsin(cos(x))' and `arccos(sin(x))' both change to `pi/2 - x'.
These simplifications are unsafe because they are valid only for
values of `x' in a certain range; outside that range, values are
folded down to the 360-degree range that the inverse trigonometric
functions always produce.

Powers of powers `(x^a)^b' are simplified to `x^(a b)' for all `a' and
`b'.  These results will be valid only in a restricted range of `x';
for example, in `(x^2)^1:2' the powers cancel to get `x', which is
valid for positive values of `x' but not for negative or complex
values.

Similarly, `sqrt(x^a)' and `sqrt(x)^a' are both simplified (possibly
unsafely) to `x^(a/2)'.

Forms like `sqrt(1 - sin(x)^2)' are simplified to, e.g., `cos(x)'.
Calc has identities of this sort for `sin', `cos', `tan', `sinh', and
`cosh'.

Arguments of square roots are partially factored to look for squared
terms that can be extracted.  For example, `sqrt(a^2 b^3 + a^3 b^2)'
simplifies to `a b sqrt(a+b)'.

The simplifications of `ln(exp(x))', `ln(e^x)', and `log10(10^x)' to
`x' are also unsafe because of problems with principal values
(although these simplifications are safe if `x' is known to be real).

Common factors are cancelled from products on both sides of an
equation, even if those factors may be zero: `a x / b x' to `a / b'.
Such factors are never cancelled from inequalities: Even `a e' is not
bold enough to reduce `a x < b x' to `a < b' (or `a > b', depending on
whether you believe `x' is positive or negative).  The `a M /' command
can be used to divide a factor out of both sides of an inequality.