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PROGRAMMING 51 The type of a data item tells us what sort of object it is. It may be numerical or non-numerical: some common types are real an ordinary number integer an integer Boolean a truth value logical a group of binary digits. Another sort of item that can occur in a program is a function. 2. Expressions, Commands and Definitions Data items and functions can be formed into expressions, which can be numerical or non-numerical depending on the component items and operators. , q, r are Boolean variables, {a + b)j{c + d) is a numerical expression p A (q V r) is a Boolean expression.

W. BARRON AND C. STRACHEY These are two functions for evaluating a square root by the well known iteration y„+i = iO* + xjyn). The second version includes the "tricks" usually incorporated in a machinecode program: it shows that functional programs are not necessarily inefficient. (iii) let CopyX [x] = Null[x] -> NIL, Atom[x] —► x, Cons[Copy\[Hd[x]l Copy\[Tl[x]]] let Copy2[x] = result of § let y - Cons[N\L NIL] Hd[y] : = Atom[Hd[x]] -> /M[JC], Co/7j2[//t/[jc]] 77[y] : = NuII[Tl[x]] -> NIL, C0/y>2[77[x]] result: = j> $ The first of these is an "orthodox" recursive function to form a copy of a list.

For example, b :=a<0-+— l,a = 0->0, 1 x : = Null[Tl[z]] -> Hd[y], Hd[Tl[y]] An alternative form of the conditional element is the conditional command. If b is a condition and C, Cl, C2 are commands (or compound commands) then the forms are if b then do C test b then do Cl or do C2 The first of these provides an "all-or-nothing" choice: if the condition b is false the command C is omitted. The second gives an "either-or" choice: 55 PROGRAMMING if b is true the command Cl is obeyed and if b is false the command Cl is obeyed.

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