# The Standard Evaluation Procedure

Here the standard procedure used by

*Mathematica* to evaluate expressions is described. This procedure is the one followed for most kinds of expression. There are, however, some kinds of expressions, such as those used to represent

*Mathematica* programs and control structures, which are evaluated in a nonstandard way.

In the standard evaluation procedure,

*Mathematica* first evaluates the head of an expression and then evaluates each element of the expressions. These elements are in general themselves expressions, to which the same evaluation procedure is recursively applied.

The three

Print functions are evaluated in turn, each printing its argument, then returning the value

Null.

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This assigns the symbol

to be

Plus.

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The head

is evaluated first, so this expression behaves just like a sum of terms.

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As soon as

*Mathematica* has evaluated the head of an expression, it sees whether the head is a symbol that has attributes. If the symbol has the attributes

Orderless,

Flat, or

Listable, then immediately after evaluating the elements of the expression

*Mathematica* performs the transformations associated with these attributes.

The next step in the standard evaluation procedure is to use definitions that

*Mathematica* knows for the expression it is evaluating.

*Mathematica* first tries to use definitions that you have made, and if there are none that apply, it tries built-in definitions.

If

*Mathematica* finds a definition that applies, it performs the corresponding transformation on the expression. The result is another expression, which must then in turn be evaluated according to the standard evaluation procedure.

• Evaluate the head of the expression. |

• Evaluate each element in turn. |

• Apply transformations associated with the attributes Orderless, Listable, and Flat. |

• Apply any definitions that you have given. |

• Apply any built-in definitions. |

• Evaluate the result. |

The standard evaluation procedure.

As discussed in

"Principles of Evaluation",

*Mathematica* follows the principle that each expression is evaluated until no further definitions apply. This means that

*Mathematica* must continue reevaluating results until it gets an expression which remains unchanged through the evaluation procedure.

Here is an example that shows how the standard evaluation procedure works on a simple expression. Assume that

.

2ax+a^2+1 | here is the original expression |

Plus[Times[2,a,x],Power[a,2],1] | this is the internal form |

Times[2,a,x] | this is evaluated first |

Times[2,7,x] | is evaluated to give 7 |

Times[14,x] | built-in definitions for Times give this result |

Power[a,2] | this is evaluated next |

Power[7,2] | here is the result after evaluating |

49 | built-in definitions for Power give this result |

Plus[Times[14,x],49,1] | here is the result after the arguments of Plus have been evaluated |

Plus[50,Times[14,x]] | built-in definitions for Plus give this result |

50+14x | the result is printed like this |

A simple example of evaluation in *Mathematica*.

*Mathematica* provides various ways to "trace" the evaluation process, as discussed in

"Tracing Evaluation". The function

Trace[expr] gives a nested list showing each subexpression generated during evaluation. (Note that the standard evaluation traverses the expression tree in a depth-first way, so that the smallest subparts of the expression appear first in the results of

Trace.)

First set

to

.

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This gives a nested list of all the subexpressions generated during the evaluation of the expression.

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The order in which

*Mathematica* applies different kinds of definitions is important. The fact that

*Mathematica* applies definitions you have given before it applies built-in definitions means that you can give definitions which override the built-in ones, as discussed in

"Modifying Built-in Functions".

This expression is evaluated using the built-in definition for

ArcSin.

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You can give your own definitions for

ArcSin. You need to remove the protection attribute first.

Your definition is used before the one that is built in.

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As discussed in

"Associating Definitions with Different Symbols", you can associate definitions with symbols either as upvalues or downvalues.

*Mathematica* always tries upvalue definitions before downvalue ones.

If you have an expression like

, there are, in general, two sets of definitions that could apply: downvalues associated with

f and upvalues associated with

g.

*Mathematica* tries the definitions associated with

g before those associated with

f.

This ordering follows the general strategy of trying specific definitions before more general ones. By applying upvalues associated with arguments before applying downvalues associated with a function,

*Mathematica* allows you to make definitions for special arguments which override the general definitions for the function with any arguments.

This defines a rule for

, to be associated with

.

This defines a rule for

, to be associated with

.

The rule associated with

is tried before the rule associated with

.

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If you remove rules associated with

, the rule associated with

is used.

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• Definitions associated with g are applied before definitions associated with f in the expression . |

The order in which definitions are applied.

Most functions such as

Plus that are built into

*Mathematica* have downvalues. There are, however, some objects in

*Mathematica* which have built-in upvalues. For example,

SeriesData objects, which represent power series, have built-in upvalues with respect to various mathematical operations.

For an expression like

, the complete sequence of definitions that are tried in the standard evaluation procedure is:

- Definitions you have given associated with g;

- Built-in definitions associated with g;

- Definitions you have given associated with f;

- Built-in definitions associated with f.

The fact that upvalues are used before downvalues is important in many situations. In a typical case, you might want to define an operation such as composition. If you give upvalues for various objects with respect to composition, these upvalues will be used whenever such objects appear. However, you can also give a general procedure for composition, to be used if no special objects are present. You can give this procedure as a downvalue for composition. Since downvalues are tried after upvalues, the general procedure will be used only if no objects with upvalues are present.

Here is a definition associated with

for composition of "

objects".

Here is a general rule for composition, associated with

.

If you compose two

objects, the rule associated with

is used.

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If you compose

objects, the general rule associated with

is used.

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In general, there can be several objects that have upvalues in a particular expression.

*Mathematica* first looks at the head of the expressio and tries any upvalues associated with it. Then it successively looks at each element of the expression, trying any upvalues that exist.

*Mathematica* performs this procedure first for upvalues that you have explicitly defined, and then for upvalues that are built-in. The procedure means that in a sequence of elements, upvalues associated with earlier elements take precedence over those associated with later elements.

This defines an upvalue for

with respect to

.

This defines an upvalue for

.

Which upvalue is used depends on which occurs first in the sequence of arguments to

.

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