The remaining three characteristics of functional programming are told quite quickly: Recursion, manipulation of lists and lazy evaluation.
Recursion
Pure functional languages support no mutable data. Instead of a loop, they use recursion. The metafunction from Pure Functions showed it already. At compile time I use recursion instead of loops. The factorial function in C++
template <int N>
struct Fac{
static int const value= N * Fac<N1>::value;
};
template <>
struct Fac<0>{
static int const value = 1;
};
can be written quite easily in Haskell:
fac 0= 1
fac n= n * fac (n1)
But, there is a small difference between the recursive factorial function in Haskell and C++. To be precise, the C++ version is not recursive. Each invocation of the general class template with the template argument N instantiates a new class template with the template argument N1. The graphic shows the process.
If you use recursion in combination with lists and pattern matching, you can create powerful functions. But, that only holds partially for C++.
Manipulation of lists
LISt Processing (LISP) is a characteristic of functional programming languages. The list is the foundation of the extremely powerful function composition in a functional language, because it is the general data structure.
The processing of lists follows a simple pattern:
 Process the first element of the list.
 Recursively process the rest of the list, reduce in each iteration by the first element.
Because list processing is so idiomatic in functional programming, there exist special names for the first element and the rest of the list: (x,xs), (head,tail) or (car,cdr).
The pattern for processing the list is directly applicable in Haskell and C++.
Firstly, the concise version of C++. The function mySum sums up the numbers from 1 to 5.
mySum [] = 0
mySum (x:xs) = x + mySum xs
mySum [1,2,3,4,5]  15
And here is the C++ version.
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template<int ...>
struct mySum;
template<>
struct mySum<>{
static const int value= 0;
};
template<int head, int ... tail>
struct mySum<head,tail...>{
static const int value= head + mySum<tail...>::value;
};
int sum= mySum<1,2,3,4,5>::value; // 15

The Haskell version is quite easy to get. Or? But the C++ version is quite heavy weight. The C++ syntax requires that the primary or also called general template must be declared. Line 4 to line 7 is the fully specialised template (metametafunction) that is used for the empty argument list. If at least on template argument is used, the partially specialised class template (line 9  12) kicks in. Let me say a few word to the three dots, the socalled ellipse. That is the reason that the class in line 14 can take an arbitrary number of arguments. The three dots in line 1 and 9 pack the template parameter pack; the three dots in line 10 and 11 unpack the function parameter pack.
Haskell and C++ apply pattern matching to use the right function.
Pattern matching
There is a subtle difference between Haskell and C++. Haskell matching strategy is first match. That's the reason, you have to define the special case first. C++ matching strategy is best match. You can use pattern matching to define the multiplication of two numbers by successively applying addition.
For the sake of elegance, C++ first.
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mult n 0 = 0
mult n 1 = n
mult n m = (mult n (m  1)) + n
mult 3 2 = (mult 3 (2  1)) + 3
= (mult 3 1 ) + 3
= 3 + 3
= 6

The lines 7  10 show the enrolled multiplication of the two numbers 3 and 2. Line 1 is applied if m == 0 holds. If m == 1 holds, line 2 is used. The general case is line 3.
C++ applies a similar strategy. The difference is that the C++ version is more verbose and that I have to define the general case at first.
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template <int N, int M>
struct Mult{
static const int value= Mult<N, M1>::value + N;
};
template <int N>
struct Mult<N, 1> {
static const int value= N;
};
template <int N>
struct Mult<N, 0> {
static const int value= 0;
};
std::cout << Mult<3, 2>::value << std::endl; // 6

Lazy evaluation
The story about lazy evaluation in C++ is quite short. That will change in C++20 with the ranges library from Eric Niebler. Lazy evaluation is the default in Haskell. Lazy evaluation means that an expression is only evaluated when needed. This strategy has two benefits.
 Lazy evaluation helps you to save time and memory.
 You can define algorithm on infinite data structures. Of course, you can only ask for a finite number of values at run time.
The following code snippet shows three impressive examples in Haskell:
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length [2+1, 3*2, 1/0, 54]  4
successor i= i: (successor (i+1))
take 5 ( successor 1 )  [1,2,3,4,5]
odds= takeWhile (< 1000) . filter odd . map (^2)
[1..]= [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 ... ControlC
odds [1..]  [1,9,25, ... , 841,961]

I can calculate in the first line the length of a list including the argument 1/0. successor in line 3 defines an infinite sequence of integers. But I only request five of them (take 5) in line 4. Therefore, all is fine. If I want to have all integers such as in line 7, I have to hit ControlC to stop the recursion. I can use the same expression [1..] as an argument for the function odds. Line 6 shows the power off function composition in Haskell. The dot (.) is the symbol for function composition. With a little exercise, you can read the function composition in line 6 from right to left: Apply to each argument the square function; let the odd elements pass and continue as long as the resulting numbers are smaller than 1000. You can the result of the application in the last list.
C++ uses by default eager evaluation. The means that, in contrary to Haskell, expressions are evaluated from the inside to the outside. C++ has short circuit evaluation. So, C++ is a little bit lazy. If the result of a logical expression is given before the whole expression was evaluated, C++ stops to evaluated the expression. Therefore, the following code snippet is valid in C++, although 1/0 is not defined.
if ( true or (1/0) ) std::cout << "short circuit evaluation" << std::endl;
What's next?
With the next post, I step into the future of C++. Fold expression in C++17 are based on variadic templates and can be used to apply the fold algorithm at compile time.
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