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### 3.1 Scalar Expressions

Using the arithmetic operations + - * / ^ (power) and parentheses, scalar expressions are composed from numbers, ordinary “scalar” variables (identifiers), array names with subscripts, operator or procedure names with arguments and statement expressions.

Examples:

x
x^3 - 2*y/(2*z^2 - df(x,z))
(p^2 + m^2)^(1/2)*log (y/m)
a(5) + b(i,q)

The symbol ** may be used as an alternative to the caret symbol (^) for forming powers, particularly in those systems that do not support a caret symbol.

Statement expressions, usually in parentheses, can also form part of a scalar expression, as in the example

w + (c:=x+y) + z .

When the algebraic value of an expression is needed, REDUCE determines it, starting with the algebraic values of the parts, roughly as follows:

Variables and operator symbols with an argument list have the algebraic values they were last assigned, or if never assigned stand for themselves. However, array elements have the algebraic values they were last assigned, or, if never assigned, are taken to be 0.

Procedures are evaluated with the values of their actual parameters.

In evaluating expressions, the standard rules of algebra are applied. Unfortunately, this algebraic evaluation of an expression is not as unambiguous as is numerical evaluation. This process is generally referred to as “simplification” in the sense that the evaluation usually but not always produces a simplified form for the expression.

There are many options available to the user for carrying out such simplification. If the user doesn’t specify any method, the default method is used. The default evaluation of an expression involves expansion of the expression and collection of like terms, ordering of the terms, evaluation of derivatives and other functions and substitution for any expressions which have values assigned or declared (see assignments and LET statements). In many cases, this is all that the user needs.

The declarations by which the user can exercise some control over the way in which the evaluation is performed are explained in other sections. For example, if a real (floating point) number is encountered during evaluation, the system will normally convert it into a ratio of two integers. If the user wants to use real arithmetic, he can effect this by the command on rounded;. Other modes for coefficient arithmetic are described elsewhere.

If an illegal action occurs during evaluation (such as division by zero) or functions are called with the wrong number of arguments, and so on, an appropriate error message is generated.

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