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12.5 The Evaluates-To Operator

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.

See TeX and LaTeX Language Modes, for the way ‘=>’ appears in TeX language output. The eqn mode gives similar treatment to ‘=>’.