# Metaprogramming

Metaprogramming is the technique of writing a computer program that operates on other programs. Systems such as compilers and program analyzers can be considered metaprograms, since they take other programs as input. The forms of metaprogramming we will discuss here are specifically concerned with generating code to be included as part of a program. In a sense, they can be considered rudimentary compilers.

# Macros and Code Generation¶

A macro is a rule that translates an input sequence into some replacement output sequence. This translation process is called macro expansion, and some languages provide macros as part of their specification. The macro facility may be implemented as a preprocessing step, where macro expansion occurs before lexical and syntactic analysis, or it may be incorporated as part of syntax analysis or a later translation step.

One of the most widely used macro systems is the C preprocessor (CPP), which is included in both C and C++ as the first step in processing a program. Preprocessor directives begin with a hash symbol and include #include, #define, #if, among others. For instance, the following defines a function-like macro to swap two items:

#define SWAP(a, b) { auto tmp = b; b = a; a = tmp; }


We can then use the macro as follows:

int main() {
int x = 3;
int y = 4;
SWAP(x, y);
cout << x << " " << y << endl;
}


Running the resulting executable will print a 4, followed by a 3.

The results of macro expansion can be obtained by passing the -E flag to g++:

$g++ -E <source>  However, the results can be quite messy if there are #includes, since that directive pulls in the code from the given file. CPP macros perform text replacement, so that the code above is equivalent to: int main() { int x = 3; int y = 4; { auto tmp = y; y = x; x = tmp; }; cout << x << " " << y << endl; }  The semicolon following the use of the SWAP macro remains, denoting an empty statement. This is a problem, however, in contexts that require a single statement, such as a conditional branch that is not enclosed by a block: if (x < y) SWAP(x, y); else cout << "no swap" << endl;  A common idiom to avoid this problem is to place the expansion code for the macro inside of a do/while: #define SWAP(a, b) do { \ auto tmp = b; \ b = a; \ a = tmp; \ } while (false)  Here, we’ve placed a backslash at the end of a line to denote that the next line should be considered a continuation of the previous one. A do/while loop syntactically ends with a semicolon, so that the semicolon in SWAP(x, y); is syntactically part of the do/while loop. Thus, the expanded code has the correct syntax: if (x < y) do { auto tmp = b; b = a; a = tmp; } while (false); else cout << "no swap" << endl;  While textual replacement is useful, it does have drawbacks, stemming from the fact that though the macros are syntactically function like, they do not behave as functions. Specifically, they do not treat arguments as their own entities, and they do not introduce a separate scope. Consider the following example: int main() { int x = 3; int y = 4; int z = 5; SWAP(x < y ? x : y, z); cout << x << " " << y << " " << z << endl; }  Running the resulting program produces the unexpected result: 3 4 3  Using g++ -E, we can see what the preprocessed code looks like. Looking only at the output for main(), we find: int main() { int x = 3; int y = 4; int z = 5; do { auto tmp = z; z = x < y ? x : y; x < y ? x : y = tmp; } while (false); cout << x << " " << y << " " << z << endl; }  Here, we’ve manually added line breaks and whitespace to make the output more readable; the preprocessor itself places the macro output on a single line. The culprit is the last generated statement: x < y ? x : y = tmp;  In C++, the conditional operator ? : and the assignment operator = have the same precedence and associate right to left, so this is equivalent to: x < y ? x : (y = tmp);  Since x < y, no assignment happens here. Thus, the value of x is unchanged. We can fix this problem by placing parentheses around each use of a macro argument: #define SWAP(a, b) do { \ auto tmp = (b); \ (b) = (a); \ (a) = tmp; \ } while (false)  This now produces the expected result, as the operations are explicitly associated by the parentheses: int main() { int x = 3; int y = 4; int z = 5; do { auto tmp = (z); (z) = (x < y ? x : y); (x < y ? x : y) = tmp; } while (false); cout << x << " " << y << " " << z << endl; }  A second problem, however, is not as immediately fixable. Consider what happens when we apply the SWAP macro to a variable named tmp: int main() { int x = 3; int tmp = 4; SWAP(tmp, x); cout << x << " " << tmp << endl; }  Running this code results in: 3 4  No swap occurs! Again, using g++ -E to examine the output, we see (modulo spacing): int main() { int x = 3; int tmp = 4; do { auto tmp = (x); (x) = (tmp); (tmp) = tmp; } while (false); cout << x << " " << tmp << endl; }  Since the temporary variable used by SWAP has the same name as an argument, the temporary captures the occurrences of the argument in the generated code. This is because the macro merely performs text substitution, which does not ensure that names get resolved to the appropriate scope. (Thus, macros do not actually use call by name, which does ensure that a name in an argument resolves to the appropriate scope.) The reliance on text replacement makes CPP a non-hygienic macro system. Other systems, such as Scheme’s, are hygienic, creating a separate scope for names introduced by a macro and ensuring that arguments are not captured. ## Scheme Macros¶ The macro system defined as part of the R5RS Scheme specification is hygienic. A macro is introduced by one of the define-syntax, let-syntax, or letrec-syntax forms, and it binds the given name to the macro. As an example, the following is a definition of let as a macro: (define-syntax let (syntax-rules () ((let ((name val) ...) body1 body2 ... ) ((lambda (name ...) body1 body2 ... ) val ... ) ) ) )  The syntax-rules from specifies the rules for the macro transformation. The first argument is a list of literals that must match between the pattern of the rule and the input. An example is the else identifier inside of a cond form. In this case, however, there are no literals. The remaining arguments to syntax-rules specify transformations. The first item in a transformation is the input pattern, and the second is the output pattern. The ... acts like a Kleene star, matching the previous item to zero or more occurrences in the input. The names that appear in an input pattern but are not in the list of literals, excepting the first item that is the macro name, are hygienic variables that match input elements. The variables can then be referenced in the output pattern to specify how to construct the output. Evaluating the expression above in the global environment binds the name let to a macro that translates to a lambda. Identifiers introduced by the body of a macro are guaranteed to avoid conflict with other identifiers, and the interpreter often renames identifiers to avoid such a conflict. Consider the following definition of a swap macro: (define-syntax swap (syntax-rules () ((swap a b) (let ((tmp b)) (set! b a) (set! a tmp) ) ) ) )  This translates a use of swap to an expression that swaps the two arguments through a temporary variable tmp. Thus: > (define x 3) > (define y 4) > (swap x y) > x 4 > y 3  However, unlike CPP macros, the tmp introduced by the swap macro is distinct from any other tmp: > (define tmp 5) > (swap x tmp) > x 5 > tmp 4  Because macros are hygienic in Scheme, we get the expected behavior. In order to support macros, the evaluation procedure of the Scheme interpreter evaluates the first item in a list, as usual. If it evaluates to a macro, then the interpreter performs macro expansion on the rest of the list without first evaluating the arguments. Any names introduced by the expansion are placed in a separate scope from other names. After expansion, the interpreter repeats the evaluation process on the result of expansion, so that if the end result is a let expression as in swap above, the expression is evaluated. A macro definition can specify multiple pattern rules. Combined with the fact that the result of expansion is evaluated, this allows a macro to be recursive, as in the following definition of let*: (define-syntax let* (syntax-rules () ((let* () body1 body2 ... ) (let () body1 body2 ... ) ) ((let* ((name1 val1) (name2 val2) ...) body1 body2 ... ) (let ((name1 val1)) (let* ((name2 val2) ...) body1 body2 ... ) ) ) ) )  There is a base-case pattern for when the let* has no bindings, in which case it translates directly into a let. There is also a recursive pattern for when there is at least one binding, in which case the let* translates into a simpler let* nested within a let. The ellipsis (...) in a macro definition is similar to a Kleene star (*) in a regular expression, denoting that the preceding item can be matched zero or more times. Thus, a let* with a single binding matches the second pattern rule above, where (name2 val2) is matched zero times. ## CPP Macros¶ We return our attention to CPP macros. Despite their non-hygienic nature, they can be very useful in tasks that involve metaprogramming. CPP allows us to use #define to define two types of macros, object-like and function-like macros. An object-lke macro is a simple text replacement, substituting one sequence of text for another. Historically, a common use was to define constants: #define PI 3.1415926535 int main() { cout << "pi = " << PI << endl; cout << "tau = " << PI * 2 << endl; }  Better practice in C++ is to define a constant using const or constexpr. A function-like macro takes arguments, as in SWAP above, and can substitute the argument text into specific locations within the replacement text. A more complex example of using function-like macros is to abstract the definition of multiple pieces of code that follow the same pattern. Consider the definition of a type to represent a complex number: struct Complex { double real; double imag; }; ostream &operator<<(ostream &os, Complex c) { return os << "(" << c.real << "+" << c.imag << "i)"; }  Suppose that in addition to the overloaded stream insertion operator above, we wish to support the arithmetic operations +, -, and *. These operations all have the same basic form: Complex operator <op>(Complex a, Complex b) { return Complex{ <expression for real>, <expression for imag> }; }  Here, we’ve used uniform initialization syntax to initialize a Complex with values for its members. We can then write a function-like macro to abstract this structure: #define COMPLEX_OP(op, real_part, imag_part) \ Complex operator op(Complex a, Complex b) { \ return Complex{ real_part, imag_part }; \ }  The macro has arguments for each piece that differs between operations, namely the operator, the expression to compute the real part, and the expression to compute the imaginary part. We can use the macro as follows to define the operations: COMPLEX_OP(+, a.real+b.real, a.imag+b.imag); COMPLEX_OP(-, a.real-b.real, a.imag-b.imag); COMPLEX_OP(*, a.real*b.real - a.imag*b.imag, a.imag*b.real + a.real*b.imag);  As with our initial SWAP implementation, the trailing semicolon is extraneous but improves readability and interaction with syntax highlighters. Running the code through the preprocessor with g++ -E, we get (modulo spacing): Complex operator +(Complex a, Complex b) { return Complex{ a.real+b.real, a.imag+b.imag }; }; Complex operator -(Complex a, Complex b) { return Complex{ a.real-b.real, a.imag-b.imag }; }; Complex operator *(Complex a, Complex b) { return Complex{ a.real*b.real - a.imag*b.imag, a.imag*b.real + a.real*b.imag }; };  We can then proceed to define operations between Complex and double values. Again, we observe that such an operation has a specific pattern: Complex operator <op>(<type1> a, <type2> b) { return <expr1> <op> <expr2>; }  Here, <exprN> is the corresponding argument converted to its Complex representation. We can abstract this using a macro: #define REAL_OP(op, typeA, typeB, argA, argB) \ Complex operator op(typeA a, typeB b) { \ return argA op argB; \ }  We can also define a macro to convert from a double to a Complex: #define CONVERT(a) \ (Complex{ a, 0 })  We can then define our operations as follows: REAL_OP(+, Complex, double, a, CONVERT(b)); REAL_OP(+, double, Complex, CONVERT(a), b); REAL_OP(-, Complex, double, a, CONVERT(b)); REAL_OP(-, double, Complex, CONVERT(a), b); REAL_OP(*, Complex, double, a, CONVERT(b)); REAL_OP(*, double, Complex, CONVERT(a), b);  Running this through the preprocessor, we get: Complex operator +(Complex a, double b) { return a + (Complex{ b, 0 }); }; Complex operator +(double a, Complex b) { return (Complex{ a, 0 }) + b; }; Complex operator -(Complex a, double b) { return a - (Complex{ b, 0 }); }; Complex operator -(double a, Complex b) { return (Complex{ a, 0 }) - b; }; Complex operator *(Complex a, double b) { return a * (Complex{ b, 0 }); }; Complex operator *(double a, Complex b) { return (Complex{ a, 0 }) * b; };  We can now use complex numbers as follows: int main() { Complex c1{ 3, 4 }; Complex c2{ -1, 2 }; double d = 0.5; cout << c1 + c2 << endl; cout << c1 - c2 << endl; cout << c1 * c2 << endl; cout << c1 + d << endl; cout << c1 - d << endl; cout << c1 * d << endl; cout << d + c1 << endl; cout << d - c1 << endl; cout << d * c1 << endl; }  This results in: (2+6i) (4+2i) (-11+2i) (3.5+4i) (2.5+4i) (1.5+2i) (3.5+4i) (-2.5+-4i) (1.5+2i)  ### Stringification and Concatenation¶ When working with macros, it can be useful to convert a macro argument to a string or to concatenate it with another token. For instance, suppose we wanted to write an interactive application that would read input from a user and perform the corresponding action. On complex numbers, the target functions may be as follows: Complex Complex_conjugate(Complex c) { return Complex{ c.real, -c.imag }; } string Complex_polar(Complex c) { return "(" + to_string(sqrt(pow(c.real, 2) + pow(c.imag, 2))) + "," + to_string(atan(c.imag / c.real)) + ")"; }  The application would compare the user input to a string representing an action, call the appropriate function, and print out the result. This has the common pattern: if (<input> == "<action>") cout << Complex_<action>(<value>) << endl;  Here, we both need a string representation of the action, as well as the ability to concatenate the Complex_ token with the action token itself. We can define a macro for this pattern as follows: #define ACTION(str, name, arg) \ if (str == #name) \ cout << Complex_ ## name(arg) << endl  The # preceding a token is the stringification operator, converting the token to a string. The ## between Complex_ and name is the token pasting operator, concatenating the tokens on either side. We can then write our application code as follows: Complex c1 { 3, 4 }; string s; while (cin >> s) { ACTION(s, conjugate, c1); ACTION(s, polar, c1); }  Running this through the preprocessor, we obtain the desired result: Complex c1 { 3, 4 }; string s; while (cin >> s) { if (s == "conjugate") cout << Complex_conjugate(c1) << endl; if (s == "polar") cout << Complex_polar(c1) << endl; }  ### The Macro Namespace¶ One pitfall of using CPP macros is that they are not contained within any particular namespace. In fact, a macro, as long as it is defined, will replace any eligible token, regardless of where the token is located. Thus, defining a macro is akin to making a particular identifier act as a reserved keyword, unable to be used by the programmer. (This is one reason why constants are usually better defined as variables qualified const or constexpr than as object-like macros.) Several conventions are used to avoid polluting the global namespace. The first is to prefix all macros with characters that are specific to the library defining them in such a way as to avoid conflict with other libraries. For instance, our complex-number macros may be prefixed with COMPLEX_ to avoid conflicting with other macros or identifiers. The second strategy is to undefine macros when they are no longer needed, using the #undef preprocessor directive. For example, at the end of our library code, we may have the following: #undef COMPLEX_OP #undef REAL_OP #undef CONVERT #undef ACTION  This frees the identifiers to be used for other purposes in later code. ## Code Generation¶ While macros allow us to generate code using the macro facilities provided by a language, there are some cases where such a facility is unavailable or otherwise insufficient for our purposes. In such a situation, it may be convenient to write a code generator in an external program, in the same language or in a different language. This technique is also called automatic programming. As an example, the R5RS Scheme specification requires implementations to provide combinations of car and cdr up to four levels deep. For instance, (caar x) should be equivalent to (car (car x)), and (caddar x) should be equivalent to (car (cdr (cdr (car x)))). Aside from car and cdr themselves, there are 28 combinations that need to be provided, which would be tedious and error-prone to write by hand. Instead, we can define the following Python script to generate a Scheme library file: import itertools def cadrify(seq): if len(seq): return '(c{0}r {1})'.format(seq[0], cadrify(seq[1:])) return 'x' def defun(seq): return '(define (c{0}r x) {1})'.format(''.join(seq), cadrify(seq)) for i in range(2, 5): for seq in itertools.product(('a', 'd'), repeat=i): print(defun(seq))  The cadrify() function is a recursive function that takes in a sequence such as ('a', 'd', 'a') and constructs a call using the first item and the recursive result of the rest of the sequence. In this example, the latter is (cdr (car x)), so the result would be (car (cdr (car x))). The base case is in which the sequence is empty, producing just x. The defun() function takes in a sequence and uses it construct the definition for the appropriate combination. It calls cadrify() to construct the body. For the sequence ('a', 'd', 'a'), the result is: (define (cadar x) (car (cdr (car x))))  Finally, the loop at the end produces all combinations of 'a' and 'd' for each length. It uses the library function itertools.product() to obtain a sequence that is the ith power of the tuple ('a', 'd'). For each combination, it calls defun() to generate the function for that combination. Running the script results in: (define (caar x) (car (car x))) (define (cadr x) (car (cdr x))) (define (cdar x) (cdr (car x))) (define (cddr x) (cdr (cdr x))) (define (caaar x) (car (car (car x)))) (define (caadr x) (car (car (cdr x)))) ... (define (cdddar x) (cdr (cdr (cdr (car x))))) (define (cddddr x) (cdr (cdr (cdr (cdr x)))))  We can place the resulting code in a standard library to be loaded by the Scheme interpreter. # Template Metaprogramming¶ Template metaprogramming is a technique that uses templates to produce source code at compile time, which is then compiled with the rest of the program’s code. It generally refers to a form of compile-time execution that takes advantage of the language’s rules for template instantiation. Template metaprogramming is most common in C++, though a handful of other languages also enable it. The key to template metaprogramming in C++ is template specialization, which allows a specialized definition to be written for instantiating a template with specific arguments. For example, consider a class template that contains a static value field that is true if the template argument is int but false otherwise. We can write the generic template as follows: template <class T> struct is_int { static const bool value = false; };  We can now define a specialization for this template when the argument is int: template <> struct is_int<int> { static const bool value = true; };  The template parameter list in a specialization contains the non-specialized parameters, if any. In the case above, there are none, so it is empty. Then after the name of the template, we provide the full set of arguments for the instantiation, in this case just int. We then provide the rest of the definition for the instantiation. Now when we use the template, the compiler uses the specialization if the template argument is compatible with the specialization, otherwise it uses the generic template: cout << is_int<double>::value << endl; cout << is_int<int>::value << endl;  This prints a 0 followed by a 1. Template specialization enables us to write code that is conditional on a template argument. Combined with recursive instantiation, this results in template instantiation being Turing complete. Templates do not encode variables that are mutable, so template metaprogramming is actually a form of functional programming. ## Pairs¶ As a more complex example, let us define pairs and lists that can be manipulated at compile time. The elements stored in these structures will be arbitrary types. Before we proceed to define pairs, we construct a reporting mechanism that allows us to examine results at compile time. We arrange to include the relevant information in an error message generated by the compiler: template <class A, int I> struct report { static_assert(I < 0, "report"); };  For simplicity, we make use of an integer template parameter, though we could encode numbers using types instead. When instantiating the report template, the static_assert raises an error if the template argument I is nonnegative. Consider the following: report<int, 5> foo;  The compiler will report an error, indicating what instantiation caused the static_assert to fail. In Clang, we get an error like the following: pair.cpp:64:3: error: static_assert failed "report" static_assert(I < 0, "report"); ^ ~~~~~ pair.cpp:67:16: note: in instantiation of template class 'report<int, 5>' requested here report<int, 5> foo; ^  Using GCC, the error is as follows: pair.cpp: In instantiation of 'struct report<int, 5>': pair.cpp:67:16: required from here main.cpp:64:3: error: static assertion failed: report static_assert(I < 0, "report"); ^  In both compilers, the relevant information is reported, which is that the arguments to the report template are int and 5. We can then define a pair template as follows: template <class First, class Second> struct pair { using car = First; using cdr = Second; };  Within the template, we define type aliases car and cdr to refer to the first and second items of the pair. Thus, pair<int, double>::car is an alias for int, while pair<int, double>::cdr is an alias for double. We can also define type aliases to extract the first and second items from a pair: template <class Pair> using car_t = typename Pair::car; template <class Pair> using cdr_t = typename Pair::cdr;  The typename keyword is required before Pair::car and Pair::cdr, since we are using a nested type whose enclosing type is dependent on a template parameter. In such a case, C++ cannot determine that we are naming a type rather than a value, so the typename keyword explicitly indicates that it is a type. Using the aliases above, car_t<pair<int, double>> is an alias for int, while cdr_t<pair<int, double>> is an alias for double. In order to represent recursive lists, we need a representation for the empty list: struct nil { };  We can now define a template to determine whether or not a list, represented either by the empty list nil or by a nil-terminated sequence of pairs, is empty. We define a generic template and then a specialization for the case of nil as the argument: template <class List> struct is_empty { static const bool value = false; }; template <> struct is_empty<nil> { static const bool value = true; };  In order to use the field value at compile time, it must be a compile-time constant, which we can arrange by making it both static and const and initializing it with a compile-time constant. With C++14, we can also define global variable templates to encode the length of a list: template <class List> const bool is_empty_v = is_empty<List>::value;  The value of is_empty_v<nil> is true, while is_empty<pair<int, nil>> is false. Then we can determine at compilation whether or not a list is empty: using x = pair<char, pair<int, pair<double, nil>>>; using y = pair<float, pair<bool, nil>>; using z = nil; report<x, is_empty_v<x>> a; report<y, is_empty_v<y>> b; report<z, is_empty_v<z>> c;  Here, we introduce type aliases for lists, which act as immutable compile-time variables. We then instantiate report with a type and whether or not it is empty. This results in the following error messages from GCC: pair.cpp: In instantiation of 'struct report<pair<char, pair<int, pair<double, nil> > >, 0>': pair.cpp:82:28: required from here pair.cpp:73:3: error: static assertion failed: report static_assert(I < 0, "report"); ^~~~~~~~~~~~~ pair.cpp: In instantiation of 'struct report<pair<float, pair<bool, nil> >, 0>': pair.cpp:83:28: required from here pair.cpp:73:3: error: static assertion failed: report pair.cpp: In instantiation of 'struct report<nil, 1>': pair.cpp:84:28: required from here pair.cpp:73:3: error: static assertion failed: report  Examining the integer argument of report, we see that the lists pair<char, pair<int, pair<double, nil>>> and pair<float, pair<bool, nil>> are not empty, but the list nil is. We can compute the length of a list using recursion: template <class List> struct length { static const int value = length<cdr_t<List>>::value + 1; }; template <> struct length<nil> { static const int value = 0; }; template <class List> const int length_v = length<List>::value;  Here, we are using a value from a recursive instantiation of the length struct. Since value is initialized with an expression consisting of an operation between compile-time constants, it is also a compile-time constant. The recursion terminates at the specialization for length<nil>, where the value member is directly initialized to 0. As with is_empty_v, we define a variable template length_v to encode the result. We can compute and report the length of the x type alias: report<x, length_v<x>> d;  The first argument to report is arbitrary, since we only care about the second argument, so we just pass x itself. We get: pair.cpp: In instantiation of 'struct report<pair<char, pair<int, pair<double, nil> > >, 3>': pair.cpp:85:26: required from here pair.cpp:73:3: error: static assertion failed: report  The relevant information is that the length is 3. We can define even more complex manipulation on lists. For instance, we can reverse a list as follows: template <class List, class SoFar> struct reverse_helper { using type = typename reverse_helper<cdr_t<List>, pair<car_t<List>, SoFar>>::type; }; template <class SoFar> struct reverse_helper<nil, SoFar> { using type = SoFar; }; template <class List> using reverse_t = typename reverse_helper<List, nil>::type;  Here, we use a helper template to perform the reversal, where the first template argument is the remaining list and the second is the reversed list so far. In each step, we compute a new partial result as pair<car_t<List>, SoFar>, adding the first item in the remaining list to the front of the previous partial result. Then cdr_t<List> is the remaining list excluding the first item. The base case of the recursion is when the remaining list is nil, in which case the final result is the same as the partial result. We accomplish this with a partial class template specialization, which allows us to specialize only some of the arguments to a class template 1. In reverse_helper, we specialize the first argument, so that any instantiation of reverse_helper where the first argument is nil will use the specialization. The specialization retains a template parameter, which is included in its parameter list. The full argument list appears after the template name, including both the specialized and unspecialized arguments. 1 C++ only allows partial specialization on class templates. Function templates may be specialized, but they cannot be partially specialized. We seed the whole computation in the reverse_t alias template with the original list and empty partial result. We apply reverse_t to x: report<reverse_t<x>, 0> e;  Here, the second argument is an arbitrary nonnegative value. We get: pair.cpp: In instantiation of 'struct report<pair<double, pair<int, pair<char, nil> > >, 0>': pair.cpp:86:27: required from here pair.cpp:73:3: error: static assertion failed: report  As a last example, we can now write a template to append two lists: template <class List1, class List2> struct append { using type = pair<car_t<List1>, typename append<cdr_t<List1>, List2>::type>; }; template <class List2> struct append<nil, List2> { using type = List2; }; template <class List1, class List2> using append_t = typename append<List1, List2>::type;  Here, the template appends the second argument to the first argument. This is accomplished by prepending the first item of the first list to the result of appending the second list to the rest of the first list. The recursion terminates when the first list is empty. Applying append_t to x and y: report<append_t<x, y>, 0> f;  We get: pair.cpp: In instantiation of 'struct report<pair<char, pair<int, pair<double, pair<float, pair<bool, nil> > > > >, 0>': pair.cpp:87:29: required from here pair.cpp:73:3: error: static assertion failed: report  ## Numerical Computations¶ Using just recursion and template specialization, we could encode numbers using a system like Church numerals. However, C++ also supports integral template parameters, so we can perform compile-time numerical computations using an integer parameter rather than just types. As an example, consider the following definition of a template to compute the factorial of the template parameter: template <int N> struct factorial { static const long long value = N * factorial<N - 1>::value; }; template <> struct factorial<0> { static const long long value = 1; };  The generic template multiplies its template argument N by the result of computing factorial on N - 1. The base case is provided by the specialization for when the argument is 0, where the factorial is 1. Here, we’ve used a long long to hold the computed value, so that larger results can be computed than can be represented by int. We define a template to report a result as follows: template <long long N> struct report { static_assert(N > 0 && N < 0, "report"); };  The condition of the static_assert is written to depend on the template parameter so that the assertion fails during instantiation, rather than before. Then if we compute the factorial of 5: report<factorial<5>::value> a;  We get: factorial.cpp: In instantiation of 'struct report<120ll>': factorial.cpp:37:34: required from here factorial.cpp:33:3: error: static assertion failed: report static_assert(N > 0 && N < 0, "report"); ^  This shows that the result is 120. We can use a macro to make our program more generic, encoding the argument to factorial as a macro that can be defined at compile time: report<factorial<NUM>::value> a;  We can even provide a default value: #ifndef NUM #define NUM 5 #endif  Then at the command line, we can specify the argument as follows: $ g++ --std=c++14 factorial.cpp -DNUM=20
factorial.cpp: In instantiation of 'struct report<2432902008176640000ll>':
factorial.cpp:27:33:   required from here
factorial.cpp:23:3: error: static assertion failed: report
static_assert(N > 0 && N < 0, "report");
^


The command-line argument -D in GCC and Clang allows us to define a macro from the command line.

Suppose we now attempt to compute the factorial of a negative number:

$g++ --std=c++14 factorial.cpp -DNUM=-1 factorial.cpp: In instantiation of 'const long long int factorial<-900>::value': factorial.cpp:23:36: recursively required from 'const long long int factorial<-2>::value' factorial.cpp:23:36: required from 'const long long int factorial<-1>::value' factorial.cpp:37:27: required from here factorial.cpp:23:36: fatal error: template instantiation depth exceeds maximum of 900 (use -ftemplate-depth= to increase the maximum) static const long long value = N * factorial<N - 1>::value; ^ compilation terminated.  We see that the recursion never reaches the base case of 0. Instead, the compiler terminates compilation when the recursion depth reaches its limit. We can attempt to add an assertion that the template argument is non-negative as follows: template <int N> struct factorial { static_assert(N >= 0, "argument to factorial must be non-negative"); static const long long value = N * factorial<N - 1>::value; };  However, this does not prevent the recursive instantiation, so that what we get is an even longer set of error messages: factorial.cpp: In instantiation of 'struct factorial<-1>': factorial.cpp:38:25: required from here factorial.cpp:23:3: error: static assertion failed: argument to factorial must be non-negative static_assert(N >= 0, "argument to factorial must be non-negative"); ^ ... factorial.cpp: In instantiation of 'struct factorial<-900>': factorial.cpp:24:36: recursively required from 'const long long int factorial<-2>::value' factorial.cpp:24:36: required from 'const long long int factorial<-1>::value' factorial.cpp:38:27: required from here factorial.cpp:23:3: error: static assertion failed: argument to factorial must be non-negative factorial.cpp: In instantiation of 'const long long int factorial<-900>::value': factorial.cpp:24:36: recursively required from 'const long long int factorial<-2>::value' factorial.cpp:24:36: required from 'const long long int factorial<-1>::value' factorial.cpp:38:27: required from here factorial.cpp:24:36: fatal error: template instantiation depth exceeds maximum of 900 (use -ftemplate-depth= to increase the maximum) static const long long value = N * factorial<N - 1>::value; ^ compilation terminated.  Here, we have removed the intermediate error messages between -1 and -900. In order to actually prevent recursive instantiation when the argument is negative, we can offload the actual recursive work to a helper template. We can then check that the argument is non-negative in factorial, converting the argument to 0 if it is negative: template <int N> struct factorial_helper { static const long long value = N * factorial_helper<N - 1>::value; }; template <> struct factorial_helper<0> { static const long long value = 1; }; template <int N> struct factorial { static_assert(N >= 0, "argument to factorial must be non-negative"); static const long long value = factorial_helper<N >= 0 ? N : 0>::value; };  The key here is that factorial only instantiates factorial_helper<0> if the argument of factorial is nonnegative. Thus, we get: $ g++ --std=c++14 factorial.cpp -DNUM=-1
factorial.cpp: In instantiation of 'struct factorial<-1>':
factorial.cpp:38:24:   required from here
factorial.cpp:17:3: error: static assertion failed: argument to factorial
must be non-negative
static_assert(N >= 0, "argument to factorial must be non-negative");
^
factorial.cpp: In instantiation of 'struct report<1ll>':
factorial.cpp:38:33:   required from here
factorial.cpp:34:3: error: static assertion failed: report
static_assert(N > 0 && N < 0, "report");
^


We no longer have an unbounded recursion. This demonstrates how we can achieve conditional compilation, even without a built-in conditional construct.

As another example, the following computes Fibonacci numbers at compile time. For simplicity, we do not implement error checking for negative arguments:

template <int N>
struct fib {
static const long long value = fib<N - 1>::value + fib<N - 2>::value;
};

template <>
struct fib<1> {
static const long long value = 1;
};

template <>
struct fib<0> {
static const long long value = 0;
};


We have two base cases, provided by separate specializations for when the argument is 0 or 1. As with factorial, we use a macro to represent the input:

report<fib<NUM>::value> a;


We can then specify the input at the command line:

$g++ --std=c++14 fib.cpp -DNUM=7 fib.cpp: In instantiation of 'struct report<13ll>': fib.cpp:26:27: required from here fib.cpp:22:3: error: static assertion failed: report static_assert(N > 0 && N < 0, "report"); ^  We can even provide the largest input for which the Fibonacci number is representable as a long long: $ g++ --std=c++14 fib.cpp -DNUM=92
fib.cpp: In instantiation of 'struct report<7540113804746346429ll>':
fib.cpp:26:27:   required from here
fib.cpp:22:3: error: static assertion failed: report
static_assert(N > 0 && N < 0, "report");
^


This computation only takes a fraction of a second, since the C++ compiler only instantiates a template once for a given set of arguments within a single translation unit. Thus, the compiler automatically performs memoization, saving the result of a single computation rather than repeating it.

While function templates can also be specialized, a function template can also be overloaded with a non-template function. In performing overload resolution, C++ prefers a non-template function over a template instantiation, as long as the parameter and return types of the template instantiation are not superior to the non-template in the given context.

As an example, consider the following function template to convert a value to a string representation:

template <class T>
string to_string(const T &item) {
std::ostringstream oss;
oss << item;
return oss.str();
}


We can make use of this template, with the compiler performing template-argument deduction, as follows:

int main() {
cout << to_string(Complex{ 3, 3.14 }) << endl;
cout << to_string(3.14) << endl;
cout << to_string(true) << endl;
}


This results in:

(3+3.14i)
3.14
1


If we then decide that the representation of a bool is undesirable, we can write a function overload as follows:

string to_string(bool item) {
return item ? "true" : "false";
}


Since this is a non-template function, C++ will prefer it to the template instantiation to_string<bool> when the argument type is bool. Thus, the same code in main() now results in:

(3+3.14i)
3.14
true


## SFINAE¶

In considering function overloads, the C++ compiler does not consider it an error if the types and expressions used in the header of a function template are unsuitable for a particular set of template arguments. This is known as substitution failure is not an error (SFINAE), and it is a powerful feature of templates in C++. Rather than producing an error in such a case, the compiler simply removes the template from the set of candidate functions to be considered in overload resolution.

As an example, suppose we wanted to modify our to_string() to use std::to_string() for the types for which the latter is defined. We can place a dependence on the existence of a suitable std::to_string() overload in the header of a new function template:

template <class T>
auto to_string(const T &item) -> decltype(std::to_string(item)) {
return std::to_string(item);
}


Here, the trailing return type is necessary so that std::to_string(item) appears in the header of the function. Then the function template will fail on substitution if there is no overload of std::to_string() such that it can be applied to a value of the template argument. For example, consider calling our to_string() on a Complex object:

cout << to_string(Complex{ 3, 3.14 }) << endl;


Our previous to_string() template is still viable, so it is considered in overload resolution. The new template we defined above, however, fails to substitute, since there is no definition of std::to_string() that can be applied to a Complex. Thus, rather than being an error, the second template is merely removed from consideration, and the call resolves to the original template.

With the second template definition, we can still call to_string() on a bool, since C++ will still prefer the non-template function. However, we run into trouble when attempting to call it on a double:

to_string.cpp:82:11: error: call to 'to_string' is ambiguous
cout << to_string(3.14) << endl;
^~~~~~~~~~
to_string.cpp:65:8: note: candidate function [with T = double]
string to_string(const T &item) {
^
to_string.cpp:72:6: note: candidate function [with T = double]
auto to_string(const T &item) -> decltype(std::to_string(item)) {
^
to_string.cpp:76:8: note: candidate function
string to_string(bool item) {
^
1 error generated.


Both templates are equally viable when the argument is of type double, so the compiler cannot disambiguate between them. The non-template overload that takes in a bool is also viable, since a double can be converted to a bool, so it is reported in the error message even though it is inferior to either template.

In order to fix this problem, we need to arrange for the first function template to be nonviable when there is a compatible overload for std::to_string(). This requires ensuring that there is a substitution failure for the template when that is the case.

## Ensuring a Substitution Failure¶

There are many tools that are used to ensure a substitution failure. Perhaps the most fundamental is the enable_if template, defined in the standard library in the <type_traits> header as of C++11. We can also define it ourselves as follows:

template <bool B, class T>
struct enable_if {
typedef T type;
};

template <class T>
struct enable_if<false, T> {
};


The generic template takes in a bool and a type and defines a member alias for the type argument. The specialization elides this alias when the bool argument is false. C++14 additionally defines enable_if_t as an alias template, as in the following:

template <bool B, class T>
using enable_if_t = typename enable_if<B, T>::type;


We can use enable_if or enable_if_t to induce a failure, as in the following definition for factorial:

template <int N>
struct factorial {
static const std::enable_if_t<N >= 0, long long> value =
N * factorial<N - 1>::value;
};


When the template argument N is negative, the enable_if instantiation has no type member, so we get an error:

In file included from factorial.cpp:1:0:
/opt/local/include/gcc5/c++/type_traits: In substitution of
'template<bool _Cond, class _Tp> using enable_if_t = typename
std::enable_if::type [with bool _Cond = false; _Tp = long long
int]':
factorial.cpp:36:52:   required from 'struct factorial<-1>'
factorial.cpp:51:25:   required from here
/opt/local/include/gcc5/c++/type_traits:2388:61: error: no type
named 'type' in 'struct std::enable_if<false, long long int>'
using enable_if_t = typename enable_if<_Cond, _Tp>::type;
^
factorial.cpp: In function 'int main()':
factorial.cpp:51:10: error: 'value' is not a member of 'factorial<-1>'
report<factorial<NUM>::value> a;
^
factorial.cpp:51:10: error: 'value' is not a member of 'factorial<-1>'
factorial.cpp:51:32: error: template argument 1 is invalid
report<factorial<NUM>::value> a;
^


This provides us another mechanism to prevent instantiation of a template with a semantically invalid argument. In this case, substitution failure is an error, since the failure did not occur in the header of a function template.

Another option we have is to rely on the fact that variadic arguments are the least preferred alternative in function-overload resolution. Thus, we can write our overloads as helper functions or function templates, with an additional argument to be considered in overload resolution:

string to_string_helper(bool item, int ignored) {
return item ? "true" : "false";
}

template <class T>
auto to_string_helper(const T &item, int ignored)
-> decltype(std::to_string(item)) {
return std::to_string(item);
}

template <class T>
string to_string_helper(const T &item, ...) {
std::ostringstream oss;
oss << item;
return oss.str();
}

template <class T>
string to_string(const T &item) {
}


Here, to_string() calls to_string_helper() with the item and a dummy integer argument. We define three overloads of to_string_helper() as before, except that the overloads for bool and types for which std::to_string() is defined take in an extra int argument. The generic overload that is viable for all types, however, uses variadic arguments. Since variadic arguments have the lowest priority in function-overload resolution, if both the generic overload and another overload are viable, the latter is chosen. Thus, the overload that uses std::to_string() is preferred when to_string_helper() is called on a double. We no longer have an ambiguity, and we get the desired result when the program is compiled and run:

(3+3.14i)
3.140000
true


As of the C++11 standard, C++ supports variadic templates, which are templates that take a variable number of arguments. Both class and function templates can be variadic, and variadic templates enable us to write variadic function overloads that are type safe, unlike C-style varargs.

As an example, consider the definition of a tuple template that encapsulates multiple items of arbitrary type. For simplicity, we implement the template to require at least one item. We can declare such a template as follows:

template <class First, class... Rest>
struct tuple;


There is a non-variadic parameter, requiring at least one argument to be provided. This is followed by a parameter pack, which accepts zero or more arguments. In this case, the ellipsis follows the class keyword, so the arguments accepted by the parameter pack are types. We can then declare a tuple as follows:

tuple<int> t1;
tuple<double, char, int> t2;


In the first instantiation, the template parameter First is associated with the argument int, while the parameter pack is empty. In the second case, the parameter First is associated with double, while the parameter pack is associated with char and int.

Within the template definition, we can use the sizeof... operator to determine the size of the parameter pack. Thus, we can compute the size of the tuple as:

static const int size = 1 + sizeof...(Rest);


Parameter packs are often processed recursively. It is natural to define a tuple itself recursively as a combination of the first data item and a smaller tuple containing all but the first:

using first_type = First;
using rest_type = tuple<Rest...>;

first_type first;
rest_type rest;


The ellipsis, when it appears to the right of a pattern containing a parameter pack, expands the pattern into comma-separated instantiations of the pattern, one per item in the parameter pack. Thus, if Rest is associated with char and int, tuple<Rest...> expands to tuple<char, int>.

In the code above, we have introduced type aliases for the type of the first data item and the type of the rest of the tuple. We then declared data members for each of these components. We can write a constructor to initialize them as follows:

tuple(First f, Rest... r) : first(f), rest(r...) {}


With First as double and Rest as above, this expands to the equivalent of:

tuple(double f, char r0, int r1) :
first(f), rest(r0, r1) {}


Both the parameter Rest... r as well as the use of the parameter r... expand, with r replaced by a unique identifier in each instantiation of the pattern.

The full template definition is as follows:

template <class First, class... Rest>
struct tuple {
static const int size = 1 + sizeof...(Rest);

using first_type = First;
using rest_type = tuple<Rest...>;

first_type first;
rest_type rest;

tuple(First f, Rest... r) : first(f), rest(r...) {}
};


Since this is a recursive definition, we need a base case to terminate the recursion. As stated above, we’ve chosen to make the base case a tuple containing one item. We can specify this base case with a specialization of the variadic template:

template <class First>
struct tuple<First> {
static const int size = 1;

using first_type = First;

first_type first;

tuple(First f) : first(f) {}
};


In order to facilitate using a tuple, we can write a function template to construct a tuple. This can then take advantage of argument deduction for function templates, which is not available for class templates prior to C++17. We write a make_tuple variadic function template as follows:

template <class... Types>
tuple<Types...> make_tuple(Types... items) {
return tuple<Types...>(items...);
}


We can now make use of this function template to construct a tuple:

tuple<int> t1 = make_tuple(3);
tuple<double, char, int> t2 = make_tuple(4.9, 'c', 3);


While we now have the ability to construct a tuple, we have not yet provided a convenient mechanism for accessing individual elements from a tuple. In order to do so, we first write a class template to contain a reference to a single element from a tuple. We declare it as follows:

template <int Index, class Tuple>
struct tuple_element;


The parameter Index is the index corresponding to the item referenced by a tuple_element, and Tuple is the type of the tuple itself. We can then write the base case as follows:

template <class Tuple>
struct tuple_element<0, Tuple> {
using type = typename Tuple::first_type;

type &item;

tuple_element(Tuple &t) : item(t.first) {}
};


The type of the element at index 0 is aliased by the first_type member of a tuple. The element itself is represented by the first data member of a tuple object. Thus, we initialize our reference to the item with the first member of the tuple argument to the constructor. We also introduce a type alias type to refer to the type of the item.

The recursive case decrements the index and passes off the computation to a tuple_element instantiated with all but the first item in a tuple:

template <int Index, class Tuple>
struct tuple_element {
using rest_type = tuple_element<Index - 1,
typename Tuple::rest_type>;
using type = typename rest_type::type;

type &item;

tuple_element(Tuple &t) : item(rest_type(t.rest).item) {}
};


The rest_type member alias of a tuple is the type representing all but the first item in the tuple. We alias rest_type in tuple_element to recursively refer to a tuple_element with a decremented index and the rest_type of the tuple. We then arrange to retrieve the item from this recursive instantiation. The constructor creates a smaller tuple_element and initializes item to refer to the item contained in the smaller tuple_element. We similarly alias type to refer to the type contained in the smaller tuple_element.

The following is an alias template for the type of a tuple element:

template <int Index, class Tuple>
using tuple_element_t = typename tuple_element<Index, Tuple>::type;


We can now write a function template to retrieve an item out of a tuple:

template <int Index, class... Types>
tuple_element_t<Index, tuple<Types...>> &get(tuple<Types...> &t) {
return tuple_element<Index, tuple<Types...>>(t).item;
}


The work is offloaded to the tuple_element class template, out of which we retrieve both the type of the element and the element itself. But since get is implemented as a function template, we can rely on argument deduction for its second template parameter:

tuple<double, char, int> t2 = make_tuple(4.9, 'c', 3);
cout << get<0>(t2) << endl;
cout << get<1>(t2) << endl;
cout << get<2>(t2) << endl;
get<0>(t2)++;
get<1>(t2)++;
get<2>(t2)++;
cout << get<0>(t2) << endl;
cout << get<1>(t2) << endl;
cout << get<2>(t2) << endl;


This results in:

4.9
c
3
5.9
d
4


The standard library provides a definition of tuple, allowing it to contain zero items, along with make_tuple() and get() in the <tuple> header.

# Example: Multidimensional Arrays¶

As an extended example of using metaprogramming to build a complex system, let’s consider the implementation of a multidimensional array library in C++. Built-in C++ arrays are very limited: they represent only a linear sequence of elements, and they do not carry any size information. Multidimensional arrays can be represented by arrays of arrays, but this representation can be cumbersome to use and can suffer from poor spatial locality. Instead, most applications linearize a multidimensional array and map a multidimensional index to a linear index. We will use this strategy, but we will abstract the translation logic behind an ADT interface.

## Points¶

We start with an abstraction for a multidimensional index, which we call a point. A point consists of a sequence of integer indices, such as $$(3, 4, 5)$$ for a three-dimensional index. We define a point template as follows:

template <int N>
struct point {
int coords[N];

int &operator[](int i) {
return coords[i];
}

const int &operator[](int i) const {
return coords[i];
}
};


The template is parameterized by the dimensionality of the point, and its data representation is an array of coordinates. We overload the index operator for both const and non-const points.

We provide a stream-insertion operator overload as follows:

template <int N>
std::ostream &operator<<(std::ostream &os, const point<N> &p) {
os << "(" << p[0];
for (int i = 1; i < N; i++) {
os << "," << p[i];
}
return os << ")";
}


In order to work with points, it is useful to have point-wise arithmetic operations on points, as well as comparison operators. For instance, the following are possible definitions of addition and equality:

template <int N>
point<N> operator+(const point<N> &a, const point<N> &b) {
point<N> result;
for (int i = 0; i < N; i++)
result[i] = a[i] + b[i];
return result;
}

template <int N>
bool operator==(const point<N> &a, const point<N> &b) {
bool result = true;
for (int i = 0; i < N; i++)
result = result && (a[i] == b[i]);
return result;
}


There is a lot of similarity between these two functions: they share the same template header, arguments, and overall body structure, with an initial value, a loop to update the value, and a return of that value. Rather than writing several arithmetic and comparison operations with this structure, we can use a function-like macro to abstract the common structure:

#define POINT_OP(op, rettype, header, action, retval)           \
template <int N>                                              \
rettype operator op(const point<N> &a, const point<N> &b) {   \
for (int i = 0; i < N; i++)                                 \
action;                                                   \
return retval;                                              \
}


Then an arithmetic operators such as + or - can be defined as follows:

POINT_OP(+, point<N>, point<N> result,
result[i] = a[i] + b[i], result);
POINT_OP(-, point<N>, point<N> result,
result[i] = a[i] - b[i], result);


These in turn are very similar, with the only difference the two occurrences of + or -. We can then abstract this structure further for arithmetic operations:

#define POINT_ARITH_OP(op)                      \
POINT_OP(op, point<N>, point<N> result,       \
result[i] = a[i] op b[i], result)


Similarly, we can abstract the structure for comparison operations:

#define POINT_COMP_OP(op, start, combiner)                      \
POINT_OP(op, bool, bool result = start,                       \
result = result combiner (a[i] op b[i]), result)


We can now use these macros to define the point operations:

POINT_ARITH_OP(+);
POINT_ARITH_OP(-);
POINT_ARITH_OP(*);
POINT_ARITH_OP(/);

POINT_COMP_OP(==, true, &&);
POINT_COMP_OP(!=, false, ||);
POINT_COMP_OP(<, true, &&);
POINT_COMP_OP(<=, true, &&);
POINT_COMP_OP(>, true, &&);
POINT_COMP_OP(>=, true, &&);


Compared to writing ten separate functions, this strategy has much less repetition.

One last operation that would be useful is to construct a point of the desired dimensionality from a sequence of coordinates, analogous to make_tuple() from the previous section. We can define a variadic function to do so as follows, giving it the name pt() for succinctness:

template <class... Is>
point<sizeof...(Is)> pt(Is... is) {
return point<sizeof...(Is)>{{ is... }};
}


We use the sizeof... operator to compute the dimensionality. The nested initializer lists are required, the outer one for the point struct itself and the inner one for initializing its coords member, since the latter is an array.

We can now perform operations on points:

cout << (pt(3, 4) + pt(1, -2)) << endl;
cout << (pt(1, 2, 3) < pt(3, 4, 5)) << endl;


This results in:

(4,2)
1


## Domains¶

The domain of an array is the set of points that it maps to elements. A domain is rectangular if the start and end index for each dimension is independent of the indices for the other dimensions. Thus, an array over a rectangular domain maps a rectangular region of space to elements.

We can represent a rectangular domain by an inclusive lower-bound point and an exclusive upper-bound point:

template <int N>
struct rectdomain {
point<N> lwb; // inclusive lower bound
point<N> upb; // exclusive upper bound

// Returns the number of points in this domain.
int size() const {
if (!(lwb < upb))
return 0;
int result = 1;
for (int i = 0; i < N; i++) {
// multiple by the span of each dimension
result *= upb[i] - lwb[i];
}
return result;
}
};


We can define an iterator over a rectangular domain as follows, writing it as a nested class within the rectdomain template:

template <int N>
struct rectdomain {
...

struct iterator {
point<N> lwb; // inclusive lower bound
point<N> upb; // inclusive upper bound
point<N> current; // current item

// Returns the current point.
point<N> operator*() const {
return current;
}

// Moves this iterator to the next point in the domain.
iterator &operator++() {
// Increment starting at the last dimension.
for (int i = N - 1; i >= 0; i--) {
current[i]++;
// If this dimension is within bounds, then we are done.
if (current[i] < upb[i])
return *this;
// Otherwise, reset this dimension to its minimum and move
// on to the previous one.
current[i] = lwb[i];
}
// We ran out of dimensions to increment, set this to an end
// iterator.
current = upb;
return *this;
}

bool operator==(const iterator &rhs) const {
return current == rhs.current;
}

bool operator!=(const iterator &rhs) const {
return !operator==(rhs);
}
};

// Return an iterator that is set to the inclusive lower-bound
// point.
iterator begin() const {
return iterator{ lwb, upb, lwb };
}

// Return an iterator that is set to the exclusive upper-bound
// point.
iterator end() const {
return iterator{ lwb, upb, upb };
}
};


The iterator keeps track of the lower and upper bounds, as well as the current point. Incrementing an iterator increments the last coordinate of the current point, and if that reaches the upper bound for that coordinate, it is set to the lower bound and the previous coordinate is incremented instead. This process is repeated as necessary, and if the first coordinate reaches its upper bound, the iterator reaches the end.

We can now use rectangular domains as follows:

for (auto p : rectdomain<3>{ pt(1, 2, 3), pt(3, 4, 5) })
cout << p << endl;


This results in:

(1,2,3)
(1,2,4)
(1,3,3)
(1,3,4)
(2,2,3)
(2,2,4)
(2,3,3)
(2,3,4)


## Arrays¶

We can now proceed to define an ADT for a multidimensional array. We can represent it with a rectangular domain and a C++ array to store the elements. We also keep track of the size of each dimension for the purposes of index computations. The following is an implementation:

template <class T, int N>
struct ndarray {
rectdomain<N> domain; // domain of this array
int sizes[N];         // cached size of each dimension
T *data;              // storage for the elements

// Constructs an array with the given domain, default initializing
// the elements.
ndarray(const rectdomain<N> &dom)
: domain(dom), data(new T[dom.size()]) {
// Compute and store sizes of each dimension.
for (int i = 0; i < N; i++) {
sizes[i] = domain.upb[i] - domain.lwb[i];
}
}

// Copy constructor does a deep copy.
ndarray(const ndarray &rhs)
: domain(rhs.domain), data(new T[domain.size()]) {
std::copy(rhs.data, rhs.data + domain.size(), data);
std::copy(rhs.sizes, rhs.sizes + N, sizes);
}

// Assignment operator does a deep copy.
ndarray &operator=(const ndarray &rhs) {
if (&rhs == this)
return *this;
delete[] data;
domain = rhs.domain;
data = new T[domain.size()];
std::copy(rhs.data, rhs.data + domain.size(), data);
std::copy(rhs.sizes, rhs.sizes + N, sizes);
return *this;
}

// Destructor deletes the underlying storage and the elements
// within.
~ndarray() {
delete[] data;
}

// Translates a multidimensional point index into a
// single-dimensional index into the storage array.
int indexof(const point<N> &index) const;

// Returns the element at the given multidimensional index.
T &operator[](const point<N> &index) {
return data[indexof(index)];
}

// Returns the element at the given multidimensional index.
const T &operator[](const point<N> &index) const {
return data[indexof(index)];
}
};


The class template is parameterized by the element type and dimensionality. A constructor takes in a rectangular domain, allocates an underlying array of the appropriate size to hold the elements, and stores the size of each dimension. The Big Three are implemented as needed. (We elide the move constructor and move assignment operator for simplicity.) We then have a function to translate a multidimensional index into a linear one, which the overloaded index operators use to obtain an element.

The indexof() function uses the combination of the input point and the size of each dimension to linearize the index. In our representation, the array is stored in row-major format, so that the last dimension is the contiguous one:

template <class T, int N>
int ndarray<T, N>::indexof(const point<N> &index) const {
int result = index[0] - domain.lwb[0];
for (int i = 1; i < N; i++) {
result = result * sizes[i-1] + (index[i] - domain.lwb[i]);
}
return result;
}


Since the value of N is a compile-time constant, this loop can be trivially unrolled by the compiler, eliminating any branching and resulting in a faster computation.

## Stencil¶

We can now use arrays to perform a stencil computation, which iteratively computes the value of a grid point based on its previous value and the previous values of its neighbors. Figure 42 is an example of a stencil update associated with Conway’s Game of Life, on a $$3 \times 3$$ grid.

Figure 42 Stencil update associated with Conway’s Game of Life.

We use two grids, one for the previous timestep and one for the current one. We use ghost regions at the edges of the grids, extending each edge by an extra point, to avoid having to do separate computations at the boundaries.

The following constructs three-dimensional grids of size $$xdim \times ydim \times zdim$$, with ghost regions:

rectdomain<3> domain{ pt(-1, -1, -1), pt(xdim+1, ydim+1, zdim+1) };
rectdomain<3> interior{ pt(0, 0, 0), pt(xdim, ydim, zdim) };
ndarray<double, 3> gridA(domain);
ndarray<double, 3> gridB(domain);


We initialize the grids as needed and then perform an iterative stencil computation as follows:

void probe(ndarray<double, 3> *gridA_ptr,
ndarray<double, 3> *gridB_ptr,
const rectdomain<3> &interior, int steps) {
for (int i = 0; i < steps; i++) {
ndarray<double, 3> &gridA = *gridA_ptr;
ndarray<double, 3> &gridB = *gridB_ptr;

for (auto p : interior) {
gridB[p] =
gridA[p + pt( 0,  0,  1)] +
gridA[p + pt( 0,  0, -1)] +
gridA[p + pt( 0,  1,  0)] +
gridA[p + pt( 0, -1,  0)] +
gridA[p + pt( 1,  0,  0)] +
gridA[p + pt(-1,  0,  0)] +
WEIGHT * gridA[p];
}

// Swap pointers
std::swap(gridA_ptr, gridB_ptr);
}
}


We make use of iteration over a rectangular domain, arithmetic over points, and using points to index into the multidimensional array. At the end of each timestep, we swap which grid is the current and which is the previous.

While this code is simple to write, it does not perform well on many compilers. The linearized iteration over the rectangular domain can prevent a compiler from optimizing the iteration order to make the best use of the memory hierarchy, such as with a polyhedral analysis. In GCC, for example, we find that a nested loop structure such as the following can be five times more efficient:

for (p[0] = interior.lwb[0]; p[0] < interior.upb[0]; p[0]++) {
for (p[1] = interior.lwb[1]; p[1] < interior.upb[1]; p[1]++) {
for (p[2] = interior.lwb[2]; p[2] < interior.upb[2]; p[2]++) {
gridB[p] =
gridA[p + pt( 0,  0,  1)] +
gridA[p + pt( 0,  0, -1)] +
gridA[p + pt( 0,  1,  0)] +
gridA[p + pt( 0, -1,  0)] +
gridA[p + pt( 1,  0,  0)] +
gridA[p + pt(-1,  0,  0)] +
WEIGHT * gridA[p];
}
}
}


This code is less simple, and it introduces a further dependency on the dimensionality of the grid, preventing us from generalizing it to an arbitrary number of dimensions.

## Nested Iteration¶

In order to solve the problem of linearized iteration, we can use metaprogramming to turn what appears to be a single loop into a nested one, making it more amenable to analysis and optimization. We start by writing a recursive template that introduces a loop nest at each level of the recursion:

template <int N>
struct rdloop {
// Performs a nested loop over the set of loop indices in [lwb,
// upb). The size of lwb and upb must be at least N. For each
// index i1, ..., iN in [lwb, upb), calls func on the point
// pt(is..., i1, ..., iN).
template <class Func, class... Indices>
static void loop(const Func &func, const int *lwb,
const int *upb, Indices... is) {
for (int i = *lwb; i < *upb; i++) {
rdloop<N-1>::loop(func, lwb+1, upb+1, is..., i);
}
}
};


We write our template as a class, since we will require a base case and would need partial function-template specialization, which is not supported by C++, to implement it purely with function templates. The class is parameterized by the dimensionality. Within the class is a single static member function template that is parameterized by a functor type and a variadic set of indices. The arguments to the function itself are a functor object, which will be applied in the innermost loop, lower and upper bounds for the remaining dimensions, and the set of indices computed so far.

The body introduces a new loop nest, using the lower and upper bounds, and recursively applies the template with one less dimension. The bound pointers are adjusted for the new dimension, and we pass the input indices along with the one for this dimension in the recursive call. Our base case, where there is only a single dimension, is then as follows:

template <>
struct rdloop<1> {
template <class Func, class... Indices>
static void loop(const Func &func, const int *lwb,
const int *upb, Indices... is) {
for (int i = *lwb; i < *upb; i++) {
func(pt(is..., i));
}
}
};


We construct a point from the collected set of indices from each dimension and then call the functor object on that point.

Now that we have a mechanism for constructing a set of nested loops, we start the recursion from a function object and domain as follows:

rdloop<N>::loop(func, domain.lwb.coords,
domain.upb.coords);


In order to actually make use of this, we provide a loop abstraction as follows:

foreach (p, interior) {
gridB[p] =
gridA[p + pt( 0,  0,  1)] +
gridA[p + pt( 0,  0, -1)] +
gridA[p + pt( 0,  1,  0)] +
gridA[p + pt( 0, -1,  0)] +
gridA[p + pt( 1,  0,  0)] +
gridA[p + pt(-1,  0,  0)] +
WEIGHT * gridA[p];
};


We have the foreach keyword, which we will define shortly, that takes in a variable name to represent a point and the domain over which to iterate. We then have a loop body that uses the point variable. A semicolon appears after the body, and it is necessary due to how foreach is defined.

The loop body looks very much like the body of a lambda function, and since we require a function object in order to build the nested structure, it is natural to consider how we can arrange for the loop body to turn into a lambda function. We need a statement in which a lambda function can appear at the end, right before the terminating semicolon, and assignment fits this structure:

<var> = [<capture>](<parameters>) {
<body>
};


Thus, we need to arrange for the foreach header to turn into the beginning of this statement:

<var> = [<capture>](<parameters>)


We would like the programmer to be able to use all local variables, so we should capture all variables by reference. The foreach also introduces a new variable for the point, so that should be in the parameter list:

<var> = [&](const point<N> &<name>)


There are several remaining things we need. First, we need to figure out the dimensionality of the point to use as the parameter. We can use decltype to do so from the domain:

<var> = [&](const decltype(<domain>.lwb) &<name>)


Second, we need a way to ensure that when this assignment happens, the nested loop structure is executed. We can do so by overloading the assignment operator of the object <var>. Finally, we also need to introduce the left-hand variable, preferably in its own scope. We can do both by introducing a dummy loop header:

#define foreach(p, dom)                                           \
for (auto _iter = (dom).iter(); !_iter.done; _iter.done = true) \
_iter = [&](const decltype((dom).lwb) &p)


In order for this to work, we need the iter() method on a domain to give us an object whose assignment operator takes in a functor. This operator would then call the functor within a nested set of loops. The object also needs a done field in order to ensure the dummy loop executes exactly one iteration. We can add the following members to the rectdomain template:

template <int N>
struct rectdomain {
...

struct fast_iter {
const rectdomain &domain; // domain over which to iterate
bool done;                // whether or not this loop has run

// Constructs a fast_iter with the given domain.
fast_iter(const rectdomain &dom)
: domain(dom), done(false) {}

// Loops over the associate domain, calling func on each point
// in the domain.
template <class Func>
fast_iter &operator=(const Func &func) {
rdloop<N>::loop(func, domain.lwb.coords,
domain.upb.coords);
return *this;
}
};

// Returns a fast_iter over this domain.
fast_iter iter() const {
return fast_iter(*this);
}
};


The assignment operator of fast_iter is a template, taking in a functor object. It then uses our nested loop generation mechanism to generate a set of nested loops and call the functor from the innermost loop, with the appropriate point as the argument.

The result is a loop that has the simplicity of a range-based for loop but, depending on the compiler, the performance of a nested set of loops. As an example, with GCC 6.2 on the author’s iMac computer, the range-based for loop takes 1.45 seconds to perform ten timesteps of the stencil above on a $$256^3$$ grid, while the nested loops and the foreach loop each take 0.28 seconds. This demonstrates the power of metaprogramming in order to extend the features of a language.