Control flow functions¶
Evaluation control¶

MaxEvalDepth
(n)¶ set the maximum evaluation depth
Use this command to set the maximum evaluation depth to
n
. The default value is 1000.The point of having a maximum evaluation depth is to catch any infinite recursion. For example, after the definition
f(x) := f(x)
, evaluating the expressionf(x)
would callf(x)
, which would callf(x)
, etc. The interpreter will halt if the maximum evaluation depth is reached. Also indirect recursion, e.g. the pair of definitionsf(x) := g(x)
andg(x) := f(x)
, will be caught.An example of an infinite recursion, caught because the maximum evaluation depth is reached:
In> f(x) := f(x) Out> True; In> f(x) Error on line 1 in file [CommandLine] Max evaluation stack depth reached. Please use MaxEvalDepth to increase the stack size as needed.
However, a long calculation may cause the maximum evaluation depth to be reached without the presence of infinite recursion. The function
MaxEvalDepth()
is meant for these cases:In> 10 # g(0) < 1; Out> True; In> 20 # g(n_IsPositiveInteger) < \ 2 * g(n1); Out> True; In> g(1001); Error on line 1 in file [CommandLine] Max evaluation stack depth reached. Please use MaxEvalDepth to increase the stack size as needed. In> MaxEvalDepth(10000); Out> True; In> g(1001); Out> 21430172143725346418968500981200036211228096234 1106721488750077674070210224987224498639675763139171 6255189345835106293650374290571384628087196915514939 7149607869135549648461970842149210124742283755908364 3060929499671638825347975351183310878921541258291423 92955373084335320859663305248773674411336138752;

Hold
(expr)¶ keep expression unevaluated
The expression
expr
is returned unevaluated. This is useful to prevent the evaluation of a certain expression in a context in which evaluation normally takes place. Example
In> Echo({ Hold(1+1), "=", 1+1 }); 1+1 = 2 Out> True;

Eval
(expr)¶ force evaluation of expression
This function explicitly requests an evaluation of the expression
expr
, and returns the result of this evaluation. Example
In> a := x; Out> x; In> x := 5; Out> 5; In> a; Out> x; In> Eval(a); Out> 5;
The variable
a
is bound tox
, andx
is bound to 5. Hence evaluatinga
will givex
. Only when an extra evaluation ofa
is requested, the value 5 is returned. Note that the behavior would be different if we had exchanged the assignments. If the assignmenta := x
were given whilex
had the value 5, the variablea
would also get the value 5 because the assignment operator:=()
evaluates the righthand side.
Conditional execution¶

If
(pred, then[, else])¶ branch point
This command implements a branch point. The predicate
pred
is evaluated, which should result in eitherTrue
orFalse
. In the first case, the expressionthen
is evaluated and returned. If the predicate yieldsFalse
, the expressionelse
(if present) is evaluated and returned. If there is noelse
branch, theIf()
expression returnsFalse
.The sign function is defined to be 1 if its argument is positive and 1 if its argument is negative. A possible implementation is:
In> mysign(x) := If (IsPositiveReal(x), 1, 1); Out> True; In> mysign(Pi); Out> 1; In> mysign(2.5); Out> 1;
Note that this will give incorrect results, if
x
cannot be numerically approximated:In> mysign(a); Out> 1;
Hence a better implementation would be:
In> mysign(_x)_IsNumber(N(x)) < If(IsPositiveReal(x), 1, 1); Out> True;
Loops¶

While
(pred) expr¶ loop while a condition is met
Keep on evaluating
expr
whilepred
evaluates toTrue
. More precisely,While()
evaluates the predicatepred
, which should evaluate to eitherTrue
orFalse
. If the result isTrue
, the expressionexpr
is evaluated and then the predicatepred
is evaluated again. If it is stillTrue
, the expressionsexpr
andpred
are again evaluated and so on untilpred
evaluates toFalse
. At that point, the loop terminates andWhile()
returnsTrue
.In particular, if
pred
immediately evaluates toFalse
, the body is never executed.While()
is the fundamental looping construct on which all other loop commands are based. It is equivalent to thewhile
command in the programming language C. Example
In> x := 0; Out> 0; In> While (x! < 10^6) [ Echo({x, x!}); x++; ]; 0 1 1 1 2 2 3 6 4 24 5 120 6 720 7 5040 8 40320 9 362880 Out> True;

Until
(pred) expr¶ loop until a condition is met
Keep on evaluating
expr
untilpred
becomesTrue
. More precisely,Until()
first evaluates the expressionbody
. Then the predicatepred
is evaluated, which should yield eitherTrue
orFalse
. In the latter case, the expressionsexpr
andpred
are again evaluated and this continues as long as “pred” isFalse
. As soon aspred
yieldsTrue
, the loop terminates andUntil()
returnsTrue
.The main difference with
While()
is thatUntil()
always evaluatesexpr
at least once, butWhile()
may not evaluate it at all. Besides, the meaning of the predicate is reversed:While()
stops ifpred
isFalse
whileUntil()
stops ifpred
isTrue
. The commandUntil(pred) expr;
is equivalent topred; While(Not pred) body;
. In fact, the implementation ofUntil()
is based on the internal commandWhile()
. TheUntil()
command can be compared to thedo ... while
construct in the programming language C. Example
In> x := 0; Out> 0; In> Until (x! > 10^6) [ Echo({x, x!}); x++; ]; 0 1 1 1 2 2 3 6 4 24 5 120 6 720 7 5040 8 40320 9 362880 Out> True;

For
(init, pred, incr) expr¶ Cstyle
for
loopThis commands implements a C style
for
loop. First of all, the expressioninit
is evaluated. Then the predicatepred
is evaluated, which should returnTrue
orFalse
. Next, the loop is executed as long as the predicate yieldsTrue
. One traversal of the loop consists of the subsequent evaluations ofexpr
,incr
, andpred
. Finally,True
is returned.This command is most often used in a form such as
For(i=1, i<=10, i++) expr
, which evaluatesexpr
withi
subsequently set to 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.The expression
For(init, pred, incr) expr
is equivalent toinit; While(pred) [expr; incr;]
. Example
In> For (i:=1, i<=10, i++) Echo({i, i!}); 1 1 2 2 3 6 4 24 5 120 6 720 7 5040 8 40320 9 362880 10 3628800 Out> True;

ForEach
(var, list) expr¶ loop over all entries in list
The expression
expr
is evaluated multiple times. The first time,var
has the value of the first element of “list”, then it gets the value of the second element and so on.ForEach()
returnsTrue
. Example
In> ForEach(i,{2,3,5,7,11}) Echo({i, i!}); 2 2 3 6 5 120 7 5040 11 39916800 Out> True;
See also

Function
() func(args)¶ 
Function
(funcname, {args}) body declare or define a function
This command can be used to define a new function with named arguments.
The number of arguments of the new function and their names are determined by the list
args
. If the ellipsis...
follows the last atom inargs
, a function with a variable number of arguments is declared (usingRuleBaseListed()
). Note that the ellipsis cannot be the only element ofargs
and must be preceded by an atom.A function with variable number of arguments can take more arguments than elements in
args
; in this case, it obtains its last argument as a list containing all extra arguments.The short form of the
Function()
call merely declares aRuleBase()
for the new function but does not define any function body. This is a convenient shorthand forRuleBase()
andRuleBaseListed()
, when definitions of the function are to be supplied by rules. If the new function has been already declared with the same number of arguments (with or without variable arguments),Function()
returns false and does nothing.The second, longer form of the
Function()
call declares a function and also defines a function body. It is equivalent to a single rule such asfuncname(_arg1, _arg2) < body
. The rule will be declared at precedence 1025. Any previous rules associated withfuncname
(with the same arity) will be discarded. More complicated functions (with more than one body) can be defined by adding more rules. Example
This will declare a new function with two or more arguments, but define no rules for it. This is equivalent to
RuleBase ("f1", {x, y, ...})
:In> Function() f1(x,y,...); Out> True; In> Function() f1(x,y); Out> False;
This defines a function
FirstOf
which returns the first element of a list. Equivalent definitions would beFirstOf(_list) < list[1]
orFirstOf(list) := list[1]
:In> Function("FirstOf", {list}) list[1]; Out> True; In> FirstOf({a,b,c}); Out> a;
The following function will print all arguments to a string:
In> Function("PrintAll",{x, ...}) If(IsList(x), PrintList(x), ToString()Write(x)); Out> True; In> PrintAll(1): Out> " 1"; In> PrintAll(1,2,3); Out> " 1 2 3";
See also
TemplateFunction()
,Rule()
,RuleBase()
,RuleBaseListed()
,:=()
,Retract()

Macro
() func(args)¶ 
Macro
(funcname, {args}) body declare or define a macro
This does the same as
Function()
, but for macros. One can define a macro easily with this function, instead of having to useDefMacroRuleBase()
. Example
The following example defines a looping function
In> Macro("myfor",{init,pred,inc,body}) [@init;While(@pred)[@body;@inc;];True;]; Out> True; In> a:=10 Out> 10;
Here this new macro
myfor
is used to loop, using a variablea
from the calling environmentIn> myfor(i:=1,i<10,i++,Echo(a*i)) 10 20 30 40 50 60 70 80 90 Out> True; In> i Out> 10;
See also

Apply
(fn, arglist)¶ apply a function to arguments
This function applies the function
fn
to the arguments inarglist
and returns the result. The first parameterfn
can either be a string containing the name of a function or a pure function. Pure functions, modeled after lambdaexpressions, have the form{varlist,body}
, wherevarlist
is the list of formal parameters. Upon application, the formal parameters are assigned the values inarglist
(the second parameter ofApply()
) and thebody
is evaluated.Another way to define a pure function is with the Lambda construct. Here, instead of passing in
{varlist,body}
, one can pass inLambda(varlist,body)
. Lambda has the advantage that its arguments are not evaluated (using lists can have undesirable effects because lists are evaluated). Lambda can be used everywhere a pure function is expected, in principle, because the functionApply()
is the only function dealing with pure functions. So all places where a pure function can be passed in will also accept Lambda.An shorthand for
Apply()
is provided by the@()
operator. Example
In> Apply("+", {5,9}); Out> 14; In> Apply({{x,y}, xy^2}, {Cos(a), Sin(a)}); Out> Cos(a)Sin(a)^2; In> Apply(Lambda({x,y}, xy^2), {Cos(a), Sin(a)}); Out> Cos(a)Sin(a)^2 In> Lambda({x,y}, xy^2) @ {Cos(a), Sin(a)} Out> Cos(a)Sin(a)^2
See also

MapArgs
(expr, fn)¶ apply a function to all toplevel arguments
Every toplevel argument in
expr
is substituted by the result of applyingfn
to this argument. Herefn
can be either the name of a function or a pure function (seeApply()
for more information on pure functions). Example
In> MapArgs(f(x,y,z),"Sin"); Out> f(Sin(x),Sin(y),Sin(z)); In> MapArgs({3,4,5,6}, {{x},x^2}); Out> {9,16,25,36};
See also

Subst
(from, to) expr¶ perform a substitution
This function substitutes every occurrence of
from
inexpr
byto
. This is a syntactical substitution: only places wherefrom
occurs as a subexpression are affected. Example
In> Subst(x, Sin(y)) x^2+x+1; Out> Sin(y)^2+Sin(y)+1; In> Subst(a+b, x) a+b+c; Out> x+c; In> Subst(b+c, x) a+b+c; Out> a+b+c;
The explanation for the last result is that the expression
a+b+c
is internally stored as(a+b)+c
. Hencea+b
is a subexpression, butb+c
is not.See also

WithValue
(var, val, expr)¶ 
WithValue
(varlist, vallist, expr) temporary assignment during an evaluation
First, the expression
val
is assigned to the variablevar
. Then, the expressionexpr
is evaluated and returned. Finally, the assignment is reversed so that the variablevar
has the same value as it had beforeWithValue()
was evaluated.The second calling sequence assigns the first element in the list of values to the first element in the list of variables, the second value to the second variable, etc.
 Example
In> WithValue(x, 3, x^2+y^2+1); Out> y^2+10; In> WithValue({x,y}, {3,2}, x^2+y^2+1); Out> 14;

expression
/:
patterns¶ local simplification rules
Sometimes you have an expression, and you want to use specific simplification rules on it that are not done by default. This can be done with the
/:
and the/::
operators. Suppose we have the expression containing things such asLn(a*b)
, and we want to change these intoLn(a)+Ln(b)
, the easiest way to do this is using the/:
operator, as follows:In> Sin(x)*Ln(a*b) Out> Sin(x)*Ln(a*b); In> % /: { Ln(_x*_y) < Ln(x)+Ln(y) } Out> Sin(x)*(Ln(a)+Ln(b));
A whole list of simplification rules can be built up in the list, and they will be applied to the expression on the left hand side of
/:
.The forms the patterns can have are one of:
pattern < replacement {pattern,replacement} {pattern,postpredicate,replacement}
Note that for these local rules,
<
should be used instead of<
which would be used in a global rule.The
/:
operator traverses an expression much asSubst()
does, that is, top down, trying to apply the rules from the beginning of the list of rules to the end of the list of rules. If the rules cannot be applied to an expression, it will try subexpressions of that expression and so on.It might be necessary sometimes to use the
/::
operator, which repeatedly applies the/:
operator until the result doesn’t change any more. Caution is required, since rules can contradict each other, which could result in an infinite loop. To detect this situation, just use/:
repeatedly on the expression. The repetitive nature should become apparent. Example
In> Sin(u)*Ln(a*b) /: {Ln(_x*_y) < Ln(x)+Ln(y)} Out> Sin(u)*(Ln(a)+Ln(b)); In> Sin(u)*Ln(a*b) /:: { a < 2, b < 3 } Out> Sin(u)*Ln(6);
See also