2.6.15 Compiling Mathematica Expressions
If you make a definition like f[x_] := x Sin[x], Mathematica will store the expression x Sin[x] in a form that can be evaluated for any x. Then when you give a particular value for x, Mathematica substitutes this value into x Sin[x], and evaluates the result. The internal code that Mathematica uses to perform this evaluation is set up to work equally well whether the value you give for x is a number, a list, an algebraic object, or any other kind of expression.
Having to take account of all these possibilities inevitably makes the evaluation process slower. However, if Mathematica could assume that x will be a machine number, then it could avoid many steps, and potentially evaluate an expression like x Sin[x] much more quickly.
Using Compile, you can construct compiled functions in Mathematica, which evaluate Mathematica expressions assuming that all the parameters which appear are numbers (or logical variables). Compile[, , ... , expr] takes an expression expr and returns a "compiled function" which evaluates this expression when given arguments , , ... .
In general, Compile creates a CompiledFunction object which contains a sequence of simple instructions for evaluating the compiled function. The instructions are chosen to be close to those found in the machine code of a typical computer, and can thus be executed quickly.
Creating compiled functions.
This defines f to be a pure function which evaluates x Sin[x] for any x.
In[1]:= f = Function[{x}, x Sin[x]]
Out[1]=
This creates a compiled function for evaluating x Sin[x].
In[2]:= fc = Compile[{x}, x Sin[x]]
Out[2]=
f and fc yield the same results, but fc runs faster when the argument you give is a number.
In[3]:= {f[2.5], fc[2.5]}
Out[3]=
Compile is useful in situations where you have to evaluate a particular numerical or logical expression many times. By taking the time to call Compile, you can get a compiled function which can be executed more quickly than an ordinary Mathematica function.
For simple expressions such as x Sin[x], there is usually little difference between the execution speed for ordinary and compiled functions. However, as the size of the expressions involved increases, the advantage of compilation also increases. For large expressions, compilation can speed up execution by a factor as large as 20.
Compilation makes the biggest difference for expressions containing a large number of simple, say arithmetic, functions. For more complicated functions, such as BesselK or Eigenvalues, most of the computation time is spent executing internal Mathematica algorithms, on which compilation has no effect.
This creates a compiled function for finding values of the tenth Legendre polynomial. The Evaluate tells Mathematica to construct the polynomial explicitly before doing compilation.
In[4]:= pc = Compile[{x}, Evaluate[LegendreP[10, x]]]
Out[4]=
This finds the value of the tenth Legendre polynomial with argument 0.4.
In[5]:= pc[0.4]
Out[5]=
This uses builtin numerical code.
In[6]:= LegendreP[10, 0.4]
Out[6]=
Even though you can use compilation to speed up numerical functions that you write, you should still try to use builtin Mathematica functions whenever possible. Builtin functions will usually run faster than any compiled Mathematica programs you can create. In addition, they typically use more extensive algorithms, with more complete control over numerical precision and so on.
You should realize that builtin Mathematica functions quite often themselves use Compile. Thus, for example, NIntegrate by default automatically uses Compile on the expression you tell it to integrate. Similarly, functions like Plot and Plot3D use Compile on the expressions you ask them to plot. Builtin functions that use Compile typically have the option Compiled. Setting Compiled > False tells the functions not to use Compile.
Specifying types for compilation.
Compile works by making assumptions about the types of objects that occur in evaluating the expression you give. The default assumption is that all variables in the expression are approximate real numbers.
Compile nevertheless also allows integers, complex numbers and logical variables (True or False), as well as arrays of numbers. You can specify the type of a particular variable by giving a pattern which matches only values that have that type. Thus, for example, you can use the pattern _Integer to specify the integer type. Similarly, you can use True  False to specify a logical variable that must be either True or False.
This compiles the expression 5 i + j with the assumption that i and j are integers.
In[7]:= Compile[{{i, _Integer}, {j, _Integer}}, 5 i + j]
Out[7]=
This yields an integer result.
In[8]:= %[8, 7]
Out[8]=
This compiles an expression that performs an operation on a matrix of integers.
In[9]:= Compile[{{m, _Integer, 2}}, Apply[Plus, Flatten[m]]]
Out[9]=
The list operations are now carried out in a compiled way, and the result is an integer.
In[10]:= %[{{1, 2, 3}, {7, 8, 9}}]
Out[10]=
The types that Compile handles correspond essentially to the types that computers typically handle at a machinecode level. Thus, for example, Compile can handle approximate real numbers that have machine precision, but it cannot handle arbitraryprecision numbers. In addition, if you specify that a particular variable is an integer, Compile generates code only for the case when the integer is of "machine size", typically between .
When the expression you ask to compile involves only standard arithmetic and logical operations, Compile can deduce the types of objects generated at every step simply from the types of the input variables. However, if you call other functions, Compile will typically not know what type of value they return. If you do not specify otherwise, Compile assumes that any other function yields an approximate real number value. You can, however, also give an explicit list of patterns, specifying what type to assume for an expression that matches a particular pattern.
This defines a function which yields an integer result when given an integer argument.
In[11]:= com[i_] := Binomial[2i, i]
This compiles x^com[i] using the assumption that com[_] is always an integer.
In[12]:= Compile[{x, {i, _Integer}}, x^com[i], {{com[_], _Integer}}]
Out[12]=
This evaluates the compiled function.
In[13]:= %[5.6, 1]
Out[13]=
The idea of Compile is to create a function which is optimized for certain types of arguments. Compile is nevertheless set up so that the functions it creates work with whatever types of arguments they are given. When the optimization cannot be used, a standard Mathematica expression is evaluated to find the value of the function.
Here is a compiled function for taking the square root of a variable.
In[14]:= sq = Compile[{x}, Sqrt[x]]
Out[14]=
If you give a real number argument, optimized code is used.
In[15]:= sq[4.5]
Out[15]=
The compiled code cannot be used, so Mathematica prints a warning, then just evaluates the original symbolic expression.
In[16]:= sq[1 + u]
Out[16]=
The compiled code generated by Compile must make assumptions not only about the types of arguments you will supply, but also about the types of all objects that arise during the execution of the code. Sometimes these types depend on the actual values of the arguments you specify. Thus, for example, Sqrt[x] yields a real number result for real x if x is not negative, but yields a complex number if x is negative.
Compile always makes a definite assumption about the type returned by a particular function. If this assumption turns out to be invalid in a particular case when the code generated by Compile is executed, then Mathematica simply abandons the compiled code in this case, and evaluates an ordinary Mathematica expression to get the result.
The compiled code does not expect a complex number, so Mathematica has to revert to explicitly evaluating the original symbolic expression.
In[17]:= sq[4.5]
Out[17]=
An important feature of Compile is that it can handle not only mathematical expressions, but also various simple Mathematica programs. Thus, for example, Compile can handle conditionals and control flow structures.
In all cases, Compile[vars, expr] holds its arguments unevaluated. This means that you can explicitly give a "program" as the expression to compile.
This creates a compiled version of a Mathematica program which implements Newton's approximation to square roots.
In[18]:= newt = Compile[ {x, {n, _Integer}}, Module[{t}, t = x; Do[t = (t + x/t)/2, {n}]; t] ]
Out[18]=
This executes the compiled code.
In[19]:= newt[2.4, 6]
Out[19]=
