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Re: Showing intermediate steps in calculations

  • To: mathgroup at
  • Subject: [mg29049] Re: [mg29019] Showing intermediate steps in calculations
  • From: "peter lindsay" <plindsay at>
  • Date: Sat, 26 May 2001 21:53:39 -0400 (EDT)
  • References: <>
  • Sender: owner-wri-mathgroup at


I attach a copy of the entry in the help files on this: see below

2.5.10 Tracing Evaluation

The standard way in which Mathematica works is to take any expression you
give as input, evaluate the expression completely, and then return the
result. When you are trying to understand what Mathematica is doing,
however, it is often worthwhile to look not just at the final result of
evaluation, but also at intermediate steps in the evaluation process.

Tracing the evaluation of expressions.

The expression 1 + 1 is evaluated immediately to 2.

Trace[1 + 1]

The 2^3 is evaluated before the addition is done.

Trace[2^3 + 4]

The evaluation of each subexpression is shown in a separate sublist.

Trace[2^3 + 4^2 + 1]

Trace[expr] gives a list which includes all the intermediate expressions
involved in the evaluation of expr. Except in rather simple cases, however,
the number of intermediate expressions generated in this way is typically
very large, and the list returned by Trace is difficult to understand.

Trace[expr, form] allows you to "filter" the expressions that Trace records,
keeping only those which match the pattern form.

Here is a recursive definition of a factorial function.

fac[n_] := n fac[n-1]; fac[1] = 1

This gives all the intermediate expressions generated in the evaluation of
fac[3]. The result is quite complicated.


This shows only intermediate expressions of the form fac[n_].

Trace[fac[3], fac[n_]]

You can specify any pattern in Trace.

Trace[fac[10], fac[n_/;n > 5]]

Trace[expr, form] effectively works by intercepting every expression that is
about to be evaluated during the evaluation of expr, and picking out those
that match the pattern form.

If you want to trace "calls" to a function like fac, you can do so simply by
telling Trace to pick out expressions of the form fac[n_]. You can also use
patterns like f[n_, 2] to pick out calls with particular argument structure.

A typical Mathematica program, however, consists not only of "function
calls" like fac[n], but also of other elements, such as assignments to
variables, control structures, and so on. All of these elements are
represented as expressions. As a result, you can use patterns in Trace to
pick out any kind of Mathematica program element. Thus, for example, you can
use a pattern like k = _ to pick out all assignments to the symbol k.

This shows the sequence of assignments made for k.

Trace[(k=2; For[i=1, i<4, i++, k = i/k]; k), k=_]

Trace[expr, form] can pick out expressions that occur at any time in the
evaluation of expr. The expressions need not, for example, appear directly
in the form of expr that you give. They may instead occur, say, during the
evaluation of functions that are called as part of the evaluation of expr.

Here is a function definition.

h[n_] := (k=n/2; Do[k = i/k, i, n]; k)

You can look for expressions generated during the evaluation of h.

Trace[h[3], k=_]

Trace allows you to monitor intermediate steps in the evaluation not only of
functions that you define, but also of some functions that are built into
Mathematica. You should realize, however, that the specific sequence of
intermediate steps followed by built­in Mathematica functions depends in
detail on their implementation and optimization in a particular version of

Some ways to use Trace.

The function Trace returns a list that represents the "history" of a
Mathematica computation. The expressions in the list are given in the order
that they were generated during the computation. In most cases, the list
returned by Trace has a nested structure, which represents the "structure"
of the computation.

The basic idea is that each sublist in the list returned by Trace represents
the "evaluation chain" for a particular Mathematica expression. The elements
of this chain correspond to different forms of the same expression. Usually,
however, the evaluation of one expression requires the evaluation of a
number of other expressions, often subexpressions. Each subsidiary
evaluation is represented by a sublist in the structure returned by Trace.

Here is a sequence of assignments.

a[1] = a[2]; a[2] = a[3]; a[3] = a[4]

This yields an evaluation chain reflecting the sequence of transformations
for a[i] used.


The successive forms generated in the simplification of y + x + y show up as
successive elements in its evaluation chain.

Trace[y + x + y]

Each argument of the function f has a separate evaluation chain, given in a

Trace[f[1 + 1, 2 + 3, 4 + 5]]

The evaluation chain for each subexpression is given in a separate sublist.

Trace[x x + y y]

Tracing the evaluation of a nested expression yields a nested list.

Trace[f[f[f[1 + 1]]]]

There are two basic ways that subsidiary evaluations can be required during
the evaluation of a Mathematica expression. The first way is that the
expression may contain subexpressions, each of which has to be evaluated.
The second way is that there may be rules for the evaluation of the
expression that involve other expressions which themselves must be
evaluated. Both kinds of subsidiary evaluations are represented by sublists
in the structure returned by Trace.

The subsidiary evaluations here come from evaluation of the arguments of f
and g.

Trace[f[g[1 + 1], 2 + 3]]

Here is a function with a condition attached.

fe[n_] := n + 1 /; EvenQ[n]

The evaluation of fe[6] involves a subsidiary evaluation associated with the


You often get nested lists when you trace the evaluation of functions that
are defined "recursively" in terms of other instances of themselves. The
reason is typically that each new instance of the function appears as a
subexpression in the expressions obtained by evaluating previous instances
of the function.

Thus, for example, with the definition fac[n_] := n fac[n-1], the evaluation
of fac[6] yields the expression 6 fac[5], which contains fac[5] as a

The successive instances of fac generated appear in successively nested

Trace[fac[6], fac[_]]

With this definition, fp[n-1] is obtained directly as the value of fp[n].

fp[n_] := fp[n - 1] /; n > 1

fp[n] never appears in a subexpression, so no sublists are generated.

Trace[fp[6], fp[_]]

Here is the recursive definition of the Fibonacci numbers.

fib[n_] := fib[n - 1] + fib[n - 2]

Here are the end conditions for the recursion.

fib[0] = fib[1] = 1

This shows all the steps in the recursive evaluation of fib[5].

Trace[fib[5], fib[_]]

Each step in the evaluation of any Mathematica expression can be thought of
as the result of applying a particular transformation rule. As discussed in
Section 2.4.10, all the rules that Mathematica knows are associated with
specific symbols or "tags". You can use Trace[expr, f] to see all the steps
in the evaluation of expr that are performed using transformation rules
associated with the symbol f. In this case, Trace gives not only the
expressions to which each rule is applied, but also the results of applying
the rules.

In general, Trace[expr, form] picks out all the steps in the evaluation of
expr where form matches either the expression about to be evaluated, or the
tag associated with the rule used.

Tracing evaluations associated with particular tags.

This shows only intermediate expressions that match fac[_].

Trace[fac[3], fac[_]]

This shows all evaluations that use transformation rules associated with the
symbol fac.

Trace[fac[3], fac]

Here is a rule for the log function.

log[x_ y_] := log[x] + log[y]

This traces the evaluation of log[a b c d], showing all transformations
associated with log.

Trace[log[a b c d], log]

Switching off tracing inside certain forms.

Trace[expr, form] allows you to trace expressions matching form generated at
any point in the evaluation of expr. Sometimes, you may want to trace only
expressions generated during certain parts of the evaluation of expr.

By setting the option TraceOn -> oform, you can specify that tracing should
be done only during the evaluation of forms which match oform. Similarly, by
setting TraceOff -> oform, you can specify that tracing should be switched
off during the evaluation of forms which match oform.

This shows all steps in the evaluation.

Trace[log[fac[2] x]]

This shows only those steps that occur during the evaluation of fac.

Trace[log[fac[2] x], TraceOn -> fac]

This shows only those steps that do not occur during the evaluation of fac.

Trace[log[fac[2] x], TraceOff -> fac]

Applying rules to expressions encountered during evaluation.

This tells Trace to return only the arguments of fib used in the evaluation
of fib[5].

Trace[fib[5], fib[n_] -> n]

A powerful aspect of the Mathematica Trace function is that the object it
returns is basically a standard Mathematica expression which you can
manipulate using other Mathematica functions. One important point to
realize, however, is that Trace wraps all expressions that appear in the
list it produces with HoldForm to prevent them from being evaluated. The
HoldForm is not displayed in standard Mathematica output format, but it is
still present in the internal structure of the expression.

This shows the expressions generated at intermediate stages in the
evaluation process.

Trace[1 + 3^2]

The expressions are wrapped with HoldForm to prevent them from evaluating.

Trace[1 + 3^2] // InputForm

In standard Mathematica output format, it is sometimes difficult to tell
which lists are associated with the structure returned by Trace, and which
are expressions being evaluated.

Trace[1 + 1, 2 + 3]

Looking at the input form resolves any ambiguities.


When you use a transformation rule in Trace, the result is evaluated before
being wrapped with HoldForm.

Trace[fac[4], fac[n_] -> n + 1]

For sophisticated computations, the list structures returned by Trace can be
quite complicated. When you use Trace[expr, form], Trace will include as
elements in the lists only those expressions which match the pattern form.
But whatever pattern you give, the nesting structure of the lists remains
the same.

This shows all occurrences of fib[_] in the evaluation of fib[3].

Trace[fib[3], fib[_]]

This shows only occurrences of fib[1], but the nesting of the lists is the
same as for fib[_].

Trace[fib[3], fib[1]]

You can set the option TraceDepth -> n to tell Trace to include only lists
nested at most n levels deep. In this way, you can often pick out the "big
steps" in a computation, without seeing the details. Note that by setting
TraceDepth or TraceOff you can avoid looking at many of the steps in a
computation, and thereby significantly speed up the operation of Trace for
that computation.

This shows only steps that appear in lists nested at most two levels deep.

Trace[fib[3], fib[_], TraceDepth->2]

Restricting the depth of tracing.

When you use Trace[expr, form], you get a list of all the expressions which
match form produced during the evaluation of expr. Sometimes it is useful to
see not only these expressions, but also the results that were obtained by
evaluating them. You can do this by setting the option TraceForward -> True
in Trace.

This shows not only expressions which match fac[_], but also the results of
evaluating those expressions.

Trace[fac[4], fac[_], TraceForward->True]

Expressions picked out using Trace[expr, form] typically lie in the middle
of an evaluation chain. By setting TraceForward -> True, you tell Trace to
include also the expression obtained at the end of the evaluation chain. If
you set TraceForward -> All, Trace will include all the expressions that
occur after the expression matching form on the evaluation chain.

With TraceForward->All, all elements on the evaluation chain after the one
that matches fac[_] are included.

Trace[fac[4], fac[_], TraceForward->All]

By setting the option TraceForward, you can effectively see what happens to
a particular form of expression during an evaluation. Sometimes, however,
you want to find out not what happens to a particular expression, but
instead how that expression was generated. You can do this by setting the
option TraceBackward. What TraceBackward does is to show you what preceded a
particular form of expression on an evaluation chain.

This shows that the number 120 came from the evaluation of fac[5] during the
evaluation of fac[10].

Trace[fac[10], 120, TraceBackward->True]

Here is the whole evaluation chain associated with the generation of the
number 120.

Trace[fac[10], 120, TraceBackward->All]

TraceForward and TraceBackward allow you to look forward and backward in a
particular evaluation chain. Sometimes, you may also want to look at the
evaluation chains within which the particular evaluation chain occurs. You
can do this using TraceAbove. If you set the option TraceAbove -> True, then
Trace will include the initial and final expressions in all the relevant
evaluation chains. With TraceAbove -> All, Trace includes all the
expressions in all these evaluation chains.

This includes the initial and final expressions in all evaluation chains
which contain the chain that contains 120.

Trace[fac[7], 120, TraceAbove->True]

This shows all the ways that fib[2] is generated during the evaluation of

Trace[fib[5], fib[2], TraceAbove->True]

Option settings for including extra steps in trace lists.

The basic way that Trace[expr, . ] works is to intercept each expression
encountered during the evaluation of expr, and then to use various criteria
to determine whether this expression should be recorded. Normally, however,
Trace intercepts expressions only after function arguments have been
evaluated. By setting TraceOriginal -> True, you can get Trace also to look
at expressions before function arguments have been evaluated.

This includes expressions which match fac[_] both before and after argument

Trace[fac[3], fac[_], TraceOriginal -> True]

The list structure produced by Trace normally includes only expressions that
constitute steps in non­trivial evaluation chains. Thus, for example,
individual symbols that evaluate to themselves are not normally included.
Nevertheless, if you set TraceOriginal -> True, then Trace looks at
absolutely every expression involved in the evaluation process, including
those that have trivial evaluation chains.

In this case, Trace includes absolutely all expressions, even those with
trivial evaluation chains.

Trace[fac[1], TraceOriginal -> True]

Additional options for Trace.

When you use Trace to study the execution of a program, there is an issue
about how local variables in the program should be treated. As discussed in
Section 2.6.3, Mathematica scoping constructs such as Module create symbols
with new names to represent local variables. Thus, even if you called a
variable x in the original code for your program, the variable may
effectively be renamed x$nnn when the program is executed.

Trace[expr, form] is set up so that by default a symbol x that appears in
form will match all symbols with names of the form x$nnn that arise in the
execution of expr. As a result, you can for example use Trace[expr, x = _]
to trace assignment to all variables, local and global, that were named x in
your original program.

Preventing the matching of local variables.

In some cases, you may want to trace only the global variable x, and not any
local variables that were originally named x. You can do this by setting the
option MatchLocalNames -> False.

This traces assignments to all variables with names of the form x$nnn.

Trace[Module[x, x = 5], x = _]

This traces assignments only to the specific global variable x.

Trace[Module[x, x = 5], x = _,

MatchLocalNames -> False]

The function Trace performs a complete computation, then returns a structure
which represents the history of the computation. Particularly in very long
computations, it is however sometimes useful to see traces of the
computation as it proceeds. The function TracePrint works essentially like
Trace, except that it prints expressions when it encounters them, rather
than saving up all of the expressions to create a list structure.

This prints expressions encountered in the evaluation of fib[3].

TracePrint[fib[3], fib[_]]


fib[3 - 1]


fib[2 - 1]


fib[2 - 2]


fib[3 - 2]


The sequence of expressions printed by TracePrint corresponds to the
sequence of expressions given in the list structure returned by Trace.
Indentation in the output from TracePrint corresponds to nesting in the list
structure from Trace. You can use the Trace options TraceOn, TraceOff and
TraceForward in TracePrint. However, since TracePrint produces output as it
goes, it cannot support the option TraceBackward. In addition, TracePrint is
set up so that TraceOriginal is effectively always set to True.

Functions for tracing evaluation.

This enters a dialog when fac[5] is encountered during the evaluation of

TraceDialog[fac[10], fac[5]]

TraceDialog::dgbgn: Entering Dialog; use Return[] to exit.

Inside the dialog you can for example find out where you are by looking at
the "stack".

Stack[ ]

This returns from the dialog, and gives the final result from the evaluation
of fac[10].

Return[ ]

TraceDialog::dgend: Exiting Dialog.

The function TraceDialog effectively allows you to stop in the middle of a
computation, and interact with the Mathematica environment that exists at
that time. You can for example find values of intermediate variables in the
computation, and even reset those values. There are however a number of
subtleties, mostly associated with pattern and module variables.

What TraceDialog does is to call the function Dialog on a sequence of
expressions. The Dialog function is discussed in detail in Section 2.13.2.
When you call Dialog, you are effectively starting a subsidiary Mathematica
session with its own sequence of input and output lines.

In general, you may need to apply arbitrary functions to the expressions you
get while tracing an evaluation. TraceScan[f, expr, . ] applies f to each
expression that arises. The expression is wrapped with HoldForm to prevent
it from evaluating.

In TraceScan[f, expr, . ], the function f is applied to expressions before
they are evaluated. TraceScan[f, expr, patt, fp] applies f before
evaluation, and fp after evaluation.

----- Original Message -----
From: "Maarten D. de Jong" <pleez.dont at rely.on.this.address>
To: mathgroup at
Subject: [mg29049] [mg29019] Showing intermediate steps in calculations

> I am writing this message on behalf of my father, who is having the
> problem. He's working through various high-school and undergraduate books
> on algebra and calculus (out of fun and interest) and he meticulously
> writes down the answers to each problem he solves. He would like to do
> on his computer, but after a bit of discussion, I discovered that any of
> the main symbolic math packages available do not solve his problem
> As far as I know, these programs take in the problem, and write out the
> answer, without showing intermediate steps and simplifications---and it is
> precisely this that my father wants to know about. Are there packages
> show these steps? Or is it just a matter of turning on an option in the
> mainstream programs?
> Thanks for any help you can offer,
> Maarten

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