The code shown here is based on an account by Thomas Hallgren (see ), extended to include factorial. 3. All solutions were written in Haskell but the algorithms easily translate to other languages. These two hand crafted functions are both much slower than the built-in factorial because Base uses some lookup table magics. The same kinds of techniques can also be used to encode behaviors more often associated with dependent types and polytypic programming, and are thus a topic of much recent interest in the Haskell community. Even if we don’t know what a factorial is, we can understand it by reading this simple code. We discussed the Fibonacci sequence, LCM and GCD. We discussed pattern matching, the Maybe Monad, filter, map and head. Ok great! Base = 0.477305071 Recursive = 517.544341882 Iterative = 491.569636915 So, the recursive factorial function is slightly slower than the iterative function. The last call returns 6, then fac(2, 3) returns 6, and finally the original call returns 6. factorial 0 acc = acc factorial n acc = factorial (n-1) $! For example, here are three different definitions of the factorial function in the language Haskell: Write a factorial function with declarative style (Haskell): factorial n = product [1..n] factorial 5 -- 120. Even a pure functional language like Haskell supports iterative solutions in the form of list comprehension. Tail Calls Consider the factorial function below: When we make the call fac(3), two recursive calls are made: fac(2, 3) and fac(1, 6). GCD was defined two ways. Haskell can use tail call optimisation to turn a recursion into a loop under the hood. Write a function which takes in an array and returns the result of adding up every item in the array: In JavaScript: factorial n = fac n 1 Where fac n acc = if n < 2 then acc else fac (n-1) (acc*n) (We'll come to what "least defined" means in a minute.) Factorial in Haskell factorial :: Integer -> Integer factorial 0 = 1 ... Iterative computation • An iterative computation is one whose execution stack is bounded by a constant, independent of the length of the computation • Iterative computation starts with an initial state S 0 2. An implementation of the factorial function can be either iterative or recursive, but the function itself isn't inherently either. Factorial in iterative and functional style public long factorial(int n) { return LongStream .rangeClosed(1, n) .reduce((a, b) -> a * b) .getAsLong(); } factorial(5) // Output: 120 It’s worth repeating that by abstracting the how part we can write more maintainable and scalable software. Note that an implementation isn't necessarily either iterative or recursive. One way took an iterative approach while the second way, Euclid’s Algorithm, used a simple recursive method. A fixed point of a function f is a value a such that f a == a.For example, 0 is a fixed point of the function (* 3) since 0 * 3 == 0.This is where the name of fix comes from: it finds the least-defined fixed point of a function. Haskell uses a lazy evaluation system which allows you define as many terms as you like, safe in the knowledge that the compiler will only allocate the ones you use in an expression. fix and fixed points []. There are quite a few cases where a recursive solution is worse than an iterative one. Iterative solution. For the two aforementioned examples that converge, this is readily seen: ( acc * n ) Note that we have used accumulator with strict evaluation in order to suppress the default laziness of Haskell computations - this code really computes new n and acc on every recursion step. Fibonacci sequence, LCM and GCD in a minute. is slightly slower than the built-in factorial because base some. 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