calc: Evaluates-To Operator
13.5 The Evaluates-To Operator
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The special algebraic symbol ‘=>’ is known as the “evaluates-to
operator”. (It will show up as an ‘evalto’ function call in other
language modes like Pascal and LaTeX.) This is a binary operator, that
is, it has a lefthand and a righthand argument, although it can be
entered with the righthand argument omitted.
A formula like ‘A => B’ is evaluated by Calc as follows: First, A is
not simplified or modified in any way. The previous value of argument B
is thrown away; the formula A is then copied and evaluated as if by the
‘=’ command according to all current modes and stored variable values,
and the result is installed as the new value of B.
For example, suppose you enter the algebraic formula ‘2 + 3 => 17’.
The number 17 is ignored, and the lefthand argument is left in its
unevaluated form; the result is the formula ‘2 + 3 => 5’.
You can enter an ‘=>’ formula either directly using algebraic entry
(in which case the righthand side may be omitted since it is going to be
replaced right away anyhow), or by using the ‘s =’ (‘calc-evalto’)
command, which takes A from the stack and replaces it with ‘A => B’.
Calc keeps track of all ‘=>’ operators on the stack, and recomputes
them whenever anything changes that might affect their values, i.e., a
mode setting or variable value. This occurs only if the ‘=>’ operator
is at the top level of the formula, or if it is part of a top-level
vector. In other words, pushing ‘2 + (a => 17)’ will change the 17 to
the actual value of ‘a’ when you enter the formula, but the result will
not be dynamically updated when ‘a’ is changed later because the ‘=>’
operator is buried inside a sum. However, a vector of ‘=>’ operators
will be recomputed, since it is convenient to push a vector like ‘[a =>,
b =>, c =>]’ on the stack to make a concise display of all the variables
in your problem. (Another way to do this would be to use ‘[a, b, c]
=>’, which provides a slightly different format of display. You can use
whichever you find easiest to read.)
The ‘m C’ (‘calc-auto-recompute’) command allows you to turn this
automatic recomputation on or off. If you turn recomputation off, you
must explicitly recompute an ‘=>’ operator on the stack in one of the
usual ways, such as by pressing ‘=’. Turning recomputation off
temporarily can save a lot of time if you will be changing several modes
or variables before you look at the ‘=>’ entries again.
Most commands are not especially useful with ‘=>’ operators as
arguments. For example, given ‘x + 2 => 17’, it won’t work to type ‘1
+’ to get ‘x + 3 => 18’. If you want to operate on the lefthand side of
the ‘=>’ operator on the top of the stack, type ‘j 1’ (that’s the digit
“one”) to select the lefthand side, execute your commands, then type ‘j
u’ to unselect.
All current modes apply when an ‘=>’ operator is computed, including
the current simplification mode. Recall that the formula
‘arcsin(sin(x))’ will not be handled by Calc’s algebraic
simplifications, but Calc’s unsafe simplifications will reduce it to
‘x’. If you enter ‘arcsin(sin(x)) =>’ normally, the result will be
‘arcsin(sin(x)) => arcsin(sin(x))’. If you change to Extended
Simplification mode, the result will be ‘arcsin(sin(x)) => x’. However,
just pressing ‘a e’ once will have no effect on ‘arcsin(sin(x)) =>
arcsin(sin(x))’, because the righthand side depends only on the lefthand
side and the current mode settings, and the lefthand side is not
affected by commands like ‘a e’.
The “let” command (‘s l’) has an interesting interaction with the
‘=>’ operator. The ‘s l’ command evaluates the second-to-top stack
entry with the top stack entry supplying a temporary value for a given
variable. As you might expect, if that stack entry is an ‘=>’ operator
its righthand side will temporarily show this value for the variable.
In fact, all ‘=>’s on the stack will be updated if they refer to that
variable. But this change is temporary in the sense that the next
command that causes Calc to look at those stack entries will make them
revert to the old variable value.
2: a => a 2: a => 17 2: a => a
1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
. . .
17 s l a <RET> p 8 <RET>
Here the ‘p 8’ command changes the current precision, thus causing
the ‘=>’ forms to be recomputed after the influence of the “let” is
gone. The ‘d <SPC>’ command (‘calc-refresh’) is a handy way to force
the ‘=>’ operators on the stack to be recomputed without any other side
effects.
Embedded mode also uses ‘=>’ operators. In Embedded mode, the
lefthand side of an ‘=>’ operator can refer to variables assigned
elsewhere in the file by ‘:=’ operators. The assignment operator ‘a :=
17’ does not actually do anything by itself. But Embedded mode
recognizes it and marks it as a sort of file-local definition of the
variable. You can enter ‘:=’ operators in Algebraic mode, or by using
the ‘s :’ (‘calc-assign’) [‘assign’] command which takes a variable and
value from the stack and replaces them with an assignment.
TeX and LaTeX Language Modes, for the way ‘=>’ appears in TeX
language output. The “eqn” mode gives similar treatment to ‘=>’.
Info Catalog
calc: Let Command
calc: Store and Recall