calc: Basic Simplifications
11.3.1 Basic Simplifications
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This section describes basic simplifications which Calc performs in many
situations. For example, both binary simplifications and algebraic
simplifications begin by performing these basic simplifications. You
can type ‘m I’ to restrict the simplifications done on the stack to
these simplifications.
The most basic simplification is the evaluation of functions. For
example, ‘2 + 3’ is evaluated to ‘5’, and ‘sqrt(9)’ is evaluated to ‘3’.
Evaluation does not occur if the arguments to a function are somehow of
the wrong type ‘tan([2,3,4])’), range (‘tan(90)’), or number
(‘tan(3,5)’), or if the function name is not recognized (‘f(5)’), or if
Symbolic mode (
Symbolic Mode) prevents evaluation (‘sqrt(2)’).
Calc simplifies (evaluates) the arguments to a function before it
simplifies the function itself. Thus ‘sqrt(5+4)’ is simplified to
‘sqrt(9)’ before the ‘sqrt’ function itself is applied. There are very
few exceptions to this rule: ‘quote’, ‘lambda’, and ‘condition’ (the
‘::’ operator) do not evaluate their arguments, ‘if’ (the ‘? :’
operator) does not evaluate all of its arguments, and ‘evalto’ does not
evaluate its lefthand argument.
Most commands apply at least these basic simplifications to all
arguments they take from the stack, perform a particular operation, then
simplify the result before pushing it back on the stack. In the common
special case of regular arithmetic commands like ‘+’ and ‘Q’ [‘sqrt’],
the arguments are simply popped from the stack and collected into a
suitable function call, which is then simplified (the arguments being
simplified first as part of the process, as described above).
Even the basic set of simplifications are too numerous to describe
completely here, but this section will describe the ones that apply to
the major arithmetic operators. This list will be rather technical in
nature, and will probably be interesting to you only if you are a
serious user of Calc’s algebra facilities.
As well as the simplifications described here, if you have stored any
rewrite rules in the variable ‘EvalRules’ then these rules will also be
applied before any of the basic simplifications. Automatic
Rewrites, for details.
And now, on with the basic simplifications:
Arithmetic operators like ‘+’ and ‘*’ always take two arguments in
Calc’s internal form. Sums and products of three or more terms are
arranged by the associative law of algebra into a left-associative form
for sums, ‘((a + b) + c) + d’, and (by default) a right-associative form
for products, ‘a * (b * (c * d))’. Formulas like ‘(a + b) + (c + d)’
are rearranged to left-associative form, though this rarely matters
since Calc’s algebra commands are designed to hide the inner structure
of sums and products as much as possible. Sums and products in their
proper associative form will be written without parentheses in the
examples below.
Sums and products are _not_ rearranged according to the commutative
law (‘a + b’ to ‘b + a’) except in a few special cases described below.
Some algebra programs always rearrange terms into a canonical order,
which enables them to see that ‘a b + b a’ can be simplified to ‘2 a b’.
If you are using Basic Simplification mode, Calc assumes you have put
the terms into the order you want and generally leaves that order alone,
with the consequence that formulas like the above will only be
simplified if you explicitly give the ‘a s’ command. Algebraic
Simplifications.
Differences ‘a - b’ are treated like sums ‘a + (-b)’ for purposes of
simplification; one of the default simplifications is to rewrite ‘a +
(-b)’ or ‘(-b) + a’, where ‘-b’ represents a “negative-looking” term,
into ‘a - b’ form. “Negative-looking” means negative numbers, negated
formulas like ‘-x’, and products or quotients in which either term is
negative-looking.
Other simplifications involving negation are ‘-(-x)’ to ‘x’; ‘-(a b)’
or ‘-(a/b)’ where either ‘a’ or ‘b’ is negative-looking, simplified by
negating that term, or else where ‘a’ or ‘b’ is any number, by negating
that number; ‘-(a + b)’ to ‘-a - b’, and ‘-(b - a)’ to ‘a - b’. (This,
and rewriting ‘(-b) + a’ to ‘a - b’, are the only cases where the order
of terms in a sum is changed by the default simplifications.)
The distributive law is used to simplify sums in some cases: ‘a x + b
x’ to ‘(a + b) x’, where ‘a’ represents a number or an implicit 1 or -1
(as in ‘x’ or ‘-x’) and similarly for ‘b’. Use the ‘a c’, ‘a f’, or ‘j
M’ commands to merge sums with non-numeric coefficients using the
distributive law.
The distributive law is only used for sums of two terms, or for
adjacent terms in a larger sum. Thus ‘a + b + b + c’ is simplified to
‘a + 2 b + c’, but ‘a + b + c + b’ is not simplified. The reason is
that comparing all terms of a sum with one another would require time
proportional to the square of the number of terms; Calc omits
potentially slow operations like this in basic simplification mode.
Finally, ‘a + 0’ and ‘0 + a’ are simplified to ‘a’. A consequence of
the above rules is that ‘0 - a’ is simplified to ‘-a’.
The products ‘1 a’ and ‘a 1’ are simplified to ‘a’; ‘(-1) a’ and ‘a
(-1)’ are simplified to ‘-a’; ‘0 a’ and ‘a 0’ are simplified to ‘0’,
except that in Matrix mode where ‘a’ is not provably scalar the result
is the generic zero matrix ‘idn(0)’, and that if ‘a’ is infinite the
result is ‘nan’.
Also, ‘(-a) b’ and ‘a (-b)’ are simplified to ‘-(a b)’, where this
occurs for negated formulas but not for regular negative numbers.
Products are commuted only to move numbers to the front: ‘a b 2’ is
commuted to ‘2 a b’.
The product ‘a (b + c)’ is distributed over the sum only if ‘a’ and
at least one of ‘b’ and ‘c’ are numbers: ‘2 (x + 3)’ goes to ‘2 x + 6’.
The formula ‘(-a) (b - c)’, where ‘-a’ is a negative number, is
rewritten to ‘a (c - b)’.
The distributive law of products and powers is used for adjacent
terms of the product: ‘x^a x^b’ goes to ‘x^(a+b)’ where ‘a’ is a number,
or an implicit 1 (as in ‘x’), or the implicit one-half of ‘sqrt(x)’, and
similarly for ‘b’. The result is written using ‘sqrt’ or ‘1/sqrt’ if
the sum of the powers is ‘1/2’ or ‘-1/2’, respectively. If the sum of
the powers is zero, the product is simplified to ‘1’ or to ‘idn(1)’ if
Matrix mode is enabled.
The product of a negative power times anything but another negative
power is changed to use division: ‘x^(-2) y’ goes to ‘y / x^2’ unless
Matrix mode is in effect and neither ‘x’ nor ‘y’ are scalar (in which
case it is considered unsafe to rearrange the order of the terms).
Finally, ‘a (b/c)’ is rewritten to ‘(a b)/c’, and also ‘(a/b) c’ is
changed to ‘(a c)/b’ unless in Matrix mode.
Simplifications for quotients are analogous to those for products.
The quotient ‘0 / x’ is simplified to ‘0’, with the same exceptions that
were noted for ‘0 x’. Likewise, ‘x / 1’ and ‘x / (-1)’ are simplified
to ‘x’ and ‘-x’, respectively.
The quotient ‘x / 0’ is left unsimplified or changed to an infinite
quantity, as directed by the current infinite mode. Infinite
Mode.
The expression ‘a / b^(-c)’ is changed to ‘a b^c’, where ‘-c’ is any
negative-looking power. Also, ‘1 / b^c’ is changed to ‘b^(-c)’ for any
power ‘c’.
Also, ‘(-a) / b’ and ‘a / (-b)’ go to ‘-(a/b)’; ‘(a/b) / c’ goes to
‘a / (b c)’; and ‘a / (b/c)’ goes to ‘(a c) / b’ unless Matrix mode
prevents this rearrangement. Similarly, ‘a / (b:c)’ is simplified to
‘(c:b) a’ for any fraction ‘b:c’.
The distributive law is applied to ‘(a + b) / c’ only if ‘c’ and at
least one of ‘a’ and ‘b’ are numbers. Quotients of powers and square
roots are distributed just as described for multiplication.
Quotients of products cancel only in the leading terms of the
numerator and denominator. In other words, ‘a x b / a y b’ is canceled
to ‘x b / y b’ but not to ‘x / y’. Once again this is because full
cancellation can be slow; use ‘a s’ to cancel all terms of the quotient.
Quotients of negative-looking values are simplified according to
‘(-a) / (-b)’ to ‘a / b’, ‘(-a) / (b - c)’ to ‘a / (c - b)’, and ‘(a -
b) / (-c)’ to ‘(b - a) / c’.
The formula ‘x^0’ is simplified to ‘1’, or to ‘idn(1)’ in Matrix
mode. The formula ‘0^x’ is simplified to ‘0’ unless ‘x’ is a negative
number, complex number or zero. If ‘x’ is negative, complex or ‘0.0’,
‘0^x’ is an infinity or an unsimplified formula according to the current
infinite mode. The expression ‘0^0’ is simplified to ‘1’.
Powers of products or quotients ‘(a b)^c’, ‘(a/b)^c’ are distributed
to ‘a^c b^c’, ‘a^c / b^c’ only if ‘c’ is an integer, or if either ‘a’ or
‘b’ are nonnegative real numbers. Powers of powers ‘(a^b)^c’ are
simplified to ‘a^(b c)’ only when ‘c’ is an integer and ‘b c’ also
evaluates to an integer. Without these restrictions these
simplifications would not be safe because of problems with principal
values. (In other words, ‘((-3)^1:2)^2’ is safe to simplify, but
‘((-3)^2)^1:2’ is not.) Declarations, for ways to inform Calc
that your variables satisfy these requirements.
As a special case of this rule, ‘sqrt(x)^n’ is simplified to
‘x^(n/2)’ only for even integers ‘n’.
If ‘a’ is known to be real, ‘b’ is an even integer, and ‘c’ is a
half- or quarter-integer, then ‘(a^b)^c’ is simplified to ‘abs(a^(b
c))’.
Also, ‘(-a)^b’ is simplified to ‘a^b’ if ‘b’ is an even integer, or
to ‘-(a^b)’ if ‘b’ is an odd integer, for any negative-looking
expression ‘-a’.
Square roots ‘sqrt(x)’ generally act like one-half powers ‘x^1:2’ for
the purposes of the above-listed simplifications.
Also, note that ‘1 / x^1:2’ is changed to ‘x^(-1:2)’, but ‘1 /
sqrt(x)’ is left alone.
Generic identity matrices (Matrix Mode) are simplified by the
following rules: ‘idn(a) + b’ to ‘a + b’ if ‘b’ is provably scalar, or
expanded out if ‘b’ is a matrix; ‘idn(a) + idn(b)’ to ‘idn(a + b)’;
‘-idn(a)’ to ‘idn(-a)’; ‘a idn(b)’ to ‘idn(a b)’ if ‘a’ is provably
scalar, or to ‘a b’ if ‘a’ is provably non-scalar; ‘idn(a) idn(b)’ to
‘idn(a b)’; analogous simplifications for quotients involving ‘idn’; and
‘idn(a)^n’ to ‘idn(a^n)’ where ‘n’ is an integer.
The ‘floor’ function and other integer truncation functions vanish if
the argument is provably integer-valued, so that ‘floor(round(x))’
simplifies to ‘round(x)’. Also, combinations of ‘float’, ‘floor’ and
its friends, and ‘ffloor’ and its friends, are simplified in appropriate
ways. Integer Truncation.
The expression ‘abs(-x)’ changes to ‘abs(x)’. The expression
‘abs(abs(x))’ changes to ‘abs(x)’; in fact, ‘abs(x)’ changes to ‘x’ or
‘-x’ if ‘x’ is provably nonnegative or nonpositive (
Declarations).
While most functions do not recognize the variable ‘i’ as an
imaginary number, the ‘arg’ function does handle the two cases ‘arg(i)’
and ‘arg(-i)’ just for convenience.
The expression ‘conj(conj(x))’ simplifies to ‘x’. Various other
expressions involving ‘conj’, ‘re’, and ‘im’ are simplified, especially
if some of the arguments are provably real or involve the constant ‘i’.
For example, ‘conj(a + b i)’ is changed to ‘conj(a) - conj(b) i’, or to
‘a - b i’ if ‘a’ and ‘b’ are known to be real.
Functions like ‘sin’ and ‘arctan’ generally don’t have any default
simplifications beyond simply evaluating the functions for suitable
numeric arguments and infinity. The algebraic simplifications described
in the next section do provide some simplifications for these functions,
though.
One important simplification that does occur is that ‘ln(e)’ is
simplified to 1, and ‘ln(e^x)’ is simplified to ‘x’ for any ‘x’. This
occurs even if you have stored a different value in the Calc variable
‘e’; but this would be a bad idea in any case if you were also using
natural logarithms!
Among the logical functions, !(A <= B) changes to A > B and so on.
Equations and inequalities where both sides are either negative-looking
or zero are simplified by negating both sides and reversing the
inequality. While it might seem reasonable to simplify ‘!!x’ to ‘x’,
this would not be valid in general because ‘!!2’ is 1, not 2.
Most other Calc functions have few if any basic simplifications
defined, aside of course from evaluation when the arguments are suitable
numbers.
Info Catalog
calc: Simplifying Formulas
calc: Simplifying Formulas
calc: Algebraic Simplifications