# J.V. Noble: A Beginners Guide to Forth

**Source:** https://galileo.phys.virginia.edu/classes/551.jvn.fall01/primer.htm

## A Beginner's Guide to Forth


by   
J.V. Noble    
  

### Contents

1. [Preliminaries](#intro)   
     
   - [Getting started](#start)   
       
     - [The structure of Forth](#struct)  
          
       - [Extending the dictionary](#dict)   
           
         - [Stacks and reverse Polish notation (RPN)](#stacks)
           1. [Manipulating the parameter stack](#param)- [Remarks on factoring](#factoring)- [The return stack and its uses](#return)- [Local variables and VALUEs](#locals)  
           - [Using memory](#memory)   
               
             - [Comparing and branching](#branch)   
                 
               - [Documenting and commenting Forth code](#comment)
                 1. [Parenthesized remarks](#parens)- [Stack comments](#stk_com)- [Drop line (\)](#backslash)- [Comment blocks](#comblok)- [Self-documenting code](#selfdoc)  
                 - [Integer arithmetic operations](#arith)   
                     
                   - [Looping and structured programming](#loops)
                     1. [Indefinite loops](#indef) - [Definite loops](#definite)- [Structured programming](#struc_prog)- [“Top-down” design](#topdown)  
                     - [CREATE ... DOES> (the pearl of Forth)](#create)
                       1. [Defining “defining” words](#def_wds) - [Run-time *vs.* compile-time actions](#runtime)- [Dimensioned data (intrinsic units)](#dimdata) - [Advanced uses of the compiler](#advanced)  
                       - [Floating point arithmetic](#fp)  
                           
                         - [Non-trivial programming example](#bin_root)  
                             
                           - [Some remarks about strings](#strings)
                             1. [Alphanumeric strings](#alph_num_str)- [Pictured Numeric Output](#num_out)  
                             - [Assmbly language programming](#code)  
                                 
                               - [Some useful references](#refs)

```
       1. Preliminaries

       Forth is an unusual computer language that has probably been applied 
       to more varied projects than any other. It is the obvious choice when 
       the project is exceptionally demanding in terms of completion sched- 
       ule, speed of execution, compactness of code, or any combination of 
       the above. 

       It has also been called “...one of the best-kept secrets in the com- 
       puting world.” This is no exaggeration: large corporations have pur-
       chased professional Forth development systems from vendors such as 
       Laboratory Microsystems, Inc., Forth, Inc. or MicroProcessor Engineer- 
       ing, Ltd. and sworn them to secrecy. 

       Some speculate (unkindly) that corporate giants prefer to hide their 
       shame at using Forth; but I believe they are actually concealing a 
       secret weapon from their rivals. Whenever Forth has competed directly 
       with a more conventional language like C it has won hands down, pro- 
       ducing smaller, faster, more reliable code in far less time. I have 
       searched for examples with the opposite outcome but have been unable 
       to find a single instance.

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2. Getting started 

       We will use Win32Forth for these illustrations. Download the file

           w32for42.exe

       and double-click on it to install on any Windows 95, -98, -NT, -ME 
       or -XP –equipped machine.


       The compressed files will then decompress themselves. They should also
       install a program group on your desktop.

       Now start Win32Forth by opening the program group and clicking on the
       appropriate icon, probably something like this: ![](win32for.gif) or ![](Win32For6.gif)


       It should respond by opening a window and writing something like

           32bit Forth for Windows 95, and NT
           Compiled: July 23rd, 1997, 5:11pm
           Version: 3.5  Build: 0008  Release Build
           Platform: Windows 95 Version: 4.0  Build: 16384
           491k bytes free
           2,719 Words in Application dictionary
           1,466 Words in System dictionary
           4,185 Words total in dictionaries
           8,293 Windows Constants available

           Loading Win32For.CFG

           *** DON'T PANIC, Press: F1 NOW! ***


       You can use UPPER or lower-case to type commands and data. Win32Forth
       is case-insensitive.

       You can also start the WinView editor by clicking on its icon in the program
       group. The features of the editor are fairly standard and self-explanatory. Make
       sure to set the preferences (Ctl-Shift-P or use the Edit drop-down menu) to
       your own liking. Among other things, WinView is written in Forth (the complete
       source is included) and is integrated with Win32Forth so that compiling errors
       pop up in the editor.

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3. The structure of Forth

       In the Win32Forth window, now type 

           BYE  <cr>

       The Win32Forth window immediately closes.


       What just happened? Forth is an interactive programming language con- 
       sisting entirely of subroutines, called words in Forth jargon.

       Interactive means you type things in at the keyboard and the machine
       responds. We will see some details of how it does this below. 

       A word is executed (interactively) by naming it. We have just seen 
       this happen: BYE is a Forth subroutine meaning “exit to the operating 
       system”. So when we typed BYE <cr> BYE was executed, and the system re- 
       turned control to Windows.


       Click on the Win32Forth icon again to re-start Forth.
       Now we will try something a little more complicated. Enter

           2 17  +  .  <cr> 19  ok 

       What happened? Forth is interpretive. A small program called the “outer 
       interpreter” continually loops, waiting for input from the keyboard or from 
       a mass storage device. The input is a sequence of text strings (words or numbers) 
       separated from each other by the standard Forth delimiter: one or more ASCII blank 
       (32decimal = 20hex) characters.

       The text strings can be interpreted in only three ways: words (subroutine names),
       numbers, or “not defined”.


       The outer interpreter tries first to look for an incoming word in the dictionary
       (a list of already-defined subroutine names). If it finds that word, the inter-
       preter executes the corresponding code.

       If no dictionary entry exists, the interpreter tries to read the input as a number.
       If the string satisfies the rules defining a number, it is converted to a number 
       in the machine's internal representation, and stored in a special memory location,
       called “the top of the stack” (TOS).


       In the above example, Forth interpreted 2 and 17 as numbers, and 
       pushed them both onto the stack. 

       "+" is a pre-defined word as is ".", so they were looked up and exe- 
       cuted. 

       "+" added 2 to 17 and left 19 on the stack. 

       The word "." (called "dot") removed 19 from the stack and displayed 
       it on the standard output device (in this case, CRT).


       The diagram below is a flow chart representing the actions performed
       by the Forth outer interpreter during interpretation.  

![](frth_int.gif)



       We might also have said

           HEX    0A  14  * . <cr>  C8 ok

       (Do you understand this? Hint: DECIMAL means “switch to decimal arith-
       metic”, whereas HEX stands for “switch to hexadecimal arithmetic”.)


       If the incoming text can neither be located in the dictionary nor in- 
       terpreted as a number, Forth issues an error message. Try it: type X <cr>
       and see 

           X
           Error: X is undefined

       or type THING <cr> and see 

           THING
           Error: THING is undefined


       Finally, here is the obligatory "Hello, World!" program. Forth lets you
       output text using the word ." as follows (we will explain in §4 below 
       what : and ; mean):

           : hi     ." Hello, World!"  ;  ok

       Now type in hi and see what happens:

           hi Hello, World! ok

       This can be elaborated with words that tab, emit carriage returns,
       display in colors, etc. but that would take us too far afield.

       (The word ." means “Display the string, following the obligatory blank space
       and terminated by the close-quote " on the standard output device.”)


       Forth belongs to the class of Threaded Interpretive Languages. This
       means it can interpret commands (subroutines or programs) typed in
       at the console, as well as create (compile) new subroutines and pro-
       grams. The compiler in a traditional language has the structure shown
       below:
![](compiler.gif)
       To compile and test a program in a traditional language such 
       as Fortran, C or Pascal, one prepares an input (source) file, submits it
       to a black box that someone else created (the compiler) and then
       runs the resulting executable file (which is generally in machine 
       language). This process can be so tedious that most program 
       development in traditional languages must be supported by
       an elaborate set of programs called the “environment”,
       consisting of integrated editors, debuggers, version control
       catalogues and the like.

       The outer interpreter/compiler of a Forth system looks like this: 
![](frth_int.gif)

       Forth has little in common with the traditional compilation method.
       Although the Forth interpreter/compiler diagrammed above looks
       complicated, it is simplicity itself compared with the contents of
       the ![](blob.gif) representing a traditional black-box compiler. 


       A continuous loop waits for input—from the keyboard, a disk file or
       whatever— and acts on it according to its nature. Input consists
       of a sequence of words and numbers. If a name is recognized it is
       executed; if it is not in the dictionary (where else would you keep
       a list of words?) Forth tries to convert it to a number and push it
       on the stack. If this is impossible, Forth aborts execution, issues an
       error message and waits for more input.

       As we shall see below, what makes Forth a compiler as well as an
       interpreter is the set of words (Forth subroutines) that, when they are
       typed in and executed, create new Forth subroutines.

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4. Extending the dictionary

       The compiler is one of Forth's most endearing features. Unlike
       most other high-level languages, the Forth compiler is part of the
       language. (LISP and its dialects also make components of the com-
       pilation mechanism available to the programmer.) That is, its com-
       ponents are Forth words available to the programmer, that can be
       used to solve his problems.

       In this section we discuss how the compiler extends the
       dictionary. As noted above, normally a Forth system awaits
       input, and interprets (and executes it). We say the system is
       normally in interpret mode.

       Forth uses special words to create new dictionary entries, i.e.,
       new words. The most important are ":" (“start a new definition”)
       and ";" (“terminate the definition”).

       Let's try this out: enter

           : *+    *  +  ;  <cr>  ok

       What happened? The word ":" was executed because it was already
       in the dictionary. The action of ":" is

         > Create a new dictionary entry named *+ and switch from
           interpret to compile mode.

         > In compile mode, the interpreter looks up words and
           —rather than executing them— installs pointers to
           their code. (If the text is a number, instead of
           pushing it on the stack, Forth builds the number
           into the dictionary space allotted for the new word,
           following special code that puts the stored number
           on the stack whenever that word is executed.)

         > The action of "*+" is thus to execute sequentially
           the previously-defined words "*" and "+".

         > The word ";" is special: when it was defined a bit
           was turned on in its dictionary entry to mark it as
           IMMEDIATE. Thus, rather than writing down the address
           of ";", the compiler executes ";" immediately.
           (That is, an IMMEDIATE word is always executed, even 
           if the system is in compile mode.) 

           What ";" does is twofold: first, it installs the code
           that returns control to the next outer level of the
           interpreter; and second, it switches back from compile
           mode to interpret mode.

       Now try out *+ :

           DECIMAL   5 6 7 *+ .  <cr>  47  ok

       This example illustrated two principles of Forth: adding a new word to
       the dictionary, and trying it out as soon as it was defined.


       The diagram below is a flow chart representing the actions performed
       by the Forth inner interpreter during compilation.       
![](inner.gif)


       Any word you have added to the dictionary can be decompiled using
       the Forth word SEE. Say

              SEE *+  <cr>

       and get

              SEE *+
              : *+    * + ;  ok

       This can be useful when trying to understand how something works.

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5. Stacks and reverse Polish notation (RPN)

       We now discuss the stack and the “reverse Polish” or “postfix” arith-
       metic based on it. (Anyone who has used a Hewlett-Packard calculator
       should be familiar with the concept.)

       Virtually all modern CPU's are designed around stacks. Forth effi-
       ciently uses its CPU by reflecting this underlying stack architecture
       in its syntax.


       But what is a stack?  As the name implies, a stack is the machine ana-
       log of a pile of cards with numbers written on them. Numbers are
       always added to the top of the pile, and removed from the top of the
       pile. The Forth input line 

           2 5 73 -16 <cr> ok 

       leaves the stack in the state 

             cell #   contents 


               0       -16        (TOS) 
               1        73        (NOS) 
               2         5 
               3         2 


       where TOS stands for “top-of-stack”, NOS for “next-on-stack”, etc. 


       We usually employ zero-based relative numbering in Forth data struct- 
       ures (such as stacks, arrays, tables, etc.) so TOS is given relative 
       #0, NOS #1, etc. 

       Suppose we followed the above input line with the line 

           + - * . <cr> xxx ok 

       what would xxx be? The operations would produce the successive stacks 

            cell#  initial     +      -       *      . 
                   stack

              0      -16      57        -52       -104 
              1       73       5          2
              2        5       2 
              3        2                                   empty  ( 104 -> CRT )
                                                           stack 

       The operation "." (TOS->display) displays -104 to the screen, leaving the 
       stack empty. That is, xxx is -104. 


       a. Manipulating the parameter stack 

       Forth systems incorporate (at least) two stacks: the parameter stack 
       and the return stack. 

       A stack-based system must provide ways to put numbers on the stack, to 
       remove them, and to rearrange their order. Forth includes standard 
       words for this purpose.  

       Putting numbers on the stack is easy: simply type the number (or in- 
       corporate it in the definition of a Forth word). 

       The word DROP removes the number from TOS and moves up all the other 
       numbers. (Since the stack usually grows downward in memory, DROP mere- 
       ly increments the pointer to TOS by 1 cell.) 

       SWAP exchanges the top 2 numbers.

       DUP duplicates the TOS into NOS. 

       ROT rotates the top 3 numbers. 


       These actions are shown below (we show what each word does to the ini- 
       tial stack) 

             cell | initial | DROP    SWAP     ROT      DUP

               0  |   -16   |  73      73        5      -16 
               1  |    73   |   5     -16      -16      -16 
               2  |     5   |   2       5       73       73 
               3  |     2   |           2        2        5 
               4  |         |                             2 


       Forth includes the words OVER, TUCK, PICK and ROLL that act as shown 
       below (note PICK and ROLL must be preceded by an integer that says 
       where on the stack an element gets PICK'ed or ROLL'ed): 

             cell | initial | OVER    TUCK    3 PICK    3 ROLL 

               0  |   -16   |  73      -16        2        2
               1  |    73   | -16       73      -16      -16 
               2  |     5   |  73      -16       73       73 
               3  |     2   |   5        5        5        5 
               4  |         |   2        2        2 

       Clearly, 0 PICK is the same as DUP, 1 PICK is a synonym for OVER, 1 ROLL 
       means SWAP and 2 ROLL means ROT. 

       The words PICK and ROLL are mainly useful for dealing with deep stacks. But
       the current trend in Forth programming is to avoid making the stack deeper
       than 3 or 4 elements. A deeper stack than that is generally considered a sign
       that the program has been insufficiently thought out and needs to be factored.



       b. Remarks on factoring
       Factoring is the process of breaking out repeated pieces of code from sub-
       routines and giving them a name. This not only shortens the overall program
       but can make the code simpler. Here is a frequently offereded illustration:

              : SUM-OF-SQUARES   ( a b -- a*a+b*b )   DUP *   SWAP   DUP *  +  ;

       has the repeated phrase DUP  *  and can be replaced profitably by

              : SQUARED   ( a -- a*a )     DUP *  ;

              : SUM-OF-SQUARES   ( a b -- a*a+b*b ) 
                   SQUARED            ( -- a b*b)
                   SWAP               ( -- b*b a)
                   SQUARED            ( -- b*b a*a)
                   +                  ( -- b*b + a*a)
              ;

       The new version of SUM-OF-SQUARES is 2 words shorter and thus easier to read.

       Before leaving this subject I would like to offer some deeper examples of factoring,
       plus a badly factored definition. To begin with the badly factored word, look at
       the file float.f in the Win32Forth main directory, and in particular at the word >float.
       This word is almost 3 pages long, which makes it virtually impossible to read or
       to maintain. 

       Here is an example of where factoring leads to great simplification. My linear equations
       solver uses Gaussian elimination with row-pivoting. By appropriate transformations the
       equations are put in the form of equations with an upper-triangular matrix. Once this is
       done the equations can be solved for xk beginning with the n'th (which only requires a
       single division to determine the unknown xn); the n-1'st can then be solved by equally
       simple algebra in terms of xn, etc. This procedure is called “back-solving” because it
       proceeds in reverse order. The word that does the all the work expects addresses of matrix
       and inhomogeneous term on the data stack:

              : }}solve     ( A{{  V{ --) 
                  initialize  triangularize  back-solve  report ;


       You will note that the words initialize, triangularize, back-solve and report were
       given names that tell you precisely what they do. They represent the major steps of
       the algorithm as described above. Of course they have to be defined before they
       can be invoked. This is why we describe the typical Forth programming style as
       “bottom-up” rather than “top-down” as with other languages.

       My final example of factoring is a routine for adaptive numerical quadrature (integrating
       a function of one variable) based on Simpson's Rule. To use it one says

              use( fn.name xa xb err )integral

       The algorithm computes the integral on an interval, then breaks it into two
       equal sub-intervals. The sub-integrals are compared with the original—if
       they agree within the tolerance the result is accumulated and the integral is
       evaluated on the remainder of the interval. If they don't agree the routine
       works on the upper sub-interval until convergence is achieved.

              : )integral    ( f: xa xb err -- I[xa,xb]) ( xt --)
                   initialize
                   BEGIN                   \ begin indefinite loop
                      subdivide               \ break last subinterval in two
                      converged?
                      IF    interpolate       \ apply Richardson interpolation
                            shrink-interval
                      THEN
                   none-left  UNTIL        \ loop until done
                   TotIntegral  F@         ( f: -- Integral)  \ leave result
              ;

       This version has been reduced to seven basic operations, not counting the
       system words BEGIN, UNTIL that set up the loop;  IF...THEN that sets
       up the branch, and F@ that puts the answer on the stack.


       c. The return stack and its uses
       We have remarked above that compilation establishes links from the 
       calling word to the previously-defined word being invoked. The linkage 
       mechanism --during execution-- uses the return stack (rstack):  the 
       address of the next word to be invoked is placed on the rstack, so 
       that when the current word is done executing, the system knows to jump 
       to the next word. (This is so in most, but not all, modern Forth imple-
       mentations. But all have a return stack, whether or not they use them
       for linking subroutines.)

       In addition to serving as a reservoir of return addresses (since words
       can be nested, the return addresses need a stack to be put on) the
       rstack is where the limits of a DO...LOOP construct are placed.

       The user can also store/retrieve to/from the rstack. This is an ex-
       ample of using a component for a purpose other than the one it was 
       designed for. Such use is discouraged for novices since it adds the 
       spice of danger to programming. See “Note of caution” below. 

       To store to the rstack we say >R , and to retrieve we say R> . The 
       word R@ copies the top of the rstack to the TOS. 


       Why use the rstack when we have a perfectly good parameter stack to 
       play with? Sometimes it becomes hard to read code that performs com-
       plex gymnastics on the stack. The rstack can reduce the complexity. 

       Alternatively, VARIABLEs —named locations— provide a place to store 
       numbers —such as intermediate results in a calculation— off the
       stack, again reducing the gymnastics. Try this:

           \ YOU DO THIS            \ EXPLANATION

           VARIABLE X <cr>  ok      \ create a named storage location X;
                                    \ X executes by leaving its address

           3 X ! <cr>  ok           \ ! ("store") expects a number and
                                    \ an address, and stores the number to
                                    \ that address

           X @  . <cr>  3 ok        \ @ ("fetch") expects an address, and
                                    \ places its contents in TOS.

       However, Forth encourages using as few named variables as possible.
       The reason: since VARIABLEs are typically global —any subroutine can
       access them— they can cause unwanted interactions among parts of a
       large program.

       Although Forth can make variables local to the subroutines that use
       them (see “headerless words” in FTR), the rstack can often replace
       local variables:

         > The rstack already exists, so it need not be defined anew.

         > When the numbers placed on it are removed, the rstack shrinks,
           reclaiming some memory.


       A note of caution: since the rstack is critical to execution we mess
       with it at our peril. If we use the rstack for temporary storage we
       must restore it to its initial state. A word that places a number on
       the rstack must remove it —using R> or RDROP (if it has been defined)
       — before exiting that word. Since DO...LOOP also uses the rstack,
       for each >R folowing DO there must be a corresponding R> or RDROP
       preceding LOOP. Neglecting these precautions will probably crash
       the system.

       RDROP is not an ANS Forth word that can be assumed predefined on any system.
       Since it is not Standard, some systems call it R>DROP (which also is not
       Standard). Here is its definition if needed:

              : RDROP   ( or R>DROP)   ( r: n -- )  R>  DROP  ;


       d. Local variables and VALUEs
       I mentioned VARIABLEs above—a VARIABLE is a subroutine whose action is to
       return the address of a named, cell-sized memory location, as in

              VARIABLE x
              x . 247496  ok   ( it doesn't have to be this address!)
              -49 x !
              x @ .  -49  ok

       A VALUE is a widely used hybrid of VARIABLE and CONSTANT (see below). We
       define and initialize a VALUE as we would a CONSTANT:

              13 VALUE  thirteen    ok

       We invoke the new VALUE just as we would a CONSTANT:

              thirteen  .  13  ok

       However, we can change a VALUE as though it were a VARIABLE:

              47  TO  thirteen  ok
              thirteen  .  47  ok

       Needless to say, the word TO also works within word definitions, replacing
       the VALUE that follows it with whatever is currently in TOS. (Note that
       it would be dangerous to follow TO with anything but a VALUE !!) VALUEs
       are part of the ANS Forth CORE EXTENSION wordset (that is, the corresponding
       code is not guaranteed to be loaded on minimal ANS-compliant systems).

       This is a good time to mention that Forth does no type-checking (unless you add
       it yourself). YOU must check that TO is followed by a VALUE and not something
       else.

       ANS Forth also includes a LOCALS EXTENSION wordset that implements named memory
       locations local to a word definition. Locals are generally dynamic in nature (that
       is, their memory is reclaimed upon exiting the word), although the Standard does
       not insist on this. A commonly used syntax is LOCALS| a b c ... |, as in this
       definition (from a line-drawing algorithm):

              : v+    ( a b c d -- a+c b+d)
                  LOCALS| d c b a |
                  a c +  b d +  ;

              2 3 4 5 v+ .S [2] 6 8  ok..  ( .S displays the stack without destroying it)

       The important things to remember are

              > the names a, b, c ... can be any Forth-acceptable strings;

              > the local names have meaning only within a word definition;

              > the locals are initialized from the stack as shown in v+ above,
                and as in the next example:

                     : test-locals  ( a b c -- )
                         LOCALS| c b a |
                         CR  ." Normal order: " a .  b .  c .
                         CR  ." Stack order:  " c .  b .  a .
                         13 TO a   14 TO b  15 TO c  \ how TO works
                         CR ." Changed: " a . b . c
                     ;  ok

                     3 4 5 test-locals
                     Normal order: 3 4 5
                     Stack order:  5 4 3
                     Changed:  13 14 15  ok

              > the locals act like VALUEs, not like VARIABLEs, as the above
                example makes clear;

              > the LOCALS EXTENSION wordset requires LOCALS| ... | to accomodate
                (at least) 8 local names.

              > LOCALS| ... | is never necessary, nor does it necessarily shorten code, as
                the example below makes clear (7 words as opposed to 6 + preamble): 

                     : v+   ( a b c d -- a+c b+d)  2>R   R>  +  SWAP  R>  +  SWAP  ;

                What it does accomplish is to reduce stack juggling and clarify the code
                in some cases.

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6. Using memory

       As we just saw, ordinary numbers are fetched from memory to
       the stack by @ ("fetch"), and stored by ! (store).

       @ expects an address on the stack and replaces that address by 
       its contents using, e.g., the phrase   X  @ 

       ! expects a number (NOS) and an address (TOS) to store it in, and 
       places the number in the memory location referred to by the address, 
       consuming both arguments in the process, as in the phrase   3 X  ! 

       Double length numbers can similarly be fetched and stored, by
       D@ and D!, if the system has these words. 

       Positive numbers that represent characters can be placed in character 
       -sized cells of memory using C@ and C!. This is convenient for operations 
       with strings of text, for example screen and keyboard I/O. 

       Of course, one cannot put numbers in memory or retrieve them,
       for that matter, without a means of allocating memory and of
       assigning labels to the memory so allocated.

       The Forth subroutines CREATE and ALLOT are the basic tools for
       setting aside memory and attaching a convenient label to it. As
       we shall see below, CREATE makes a new dictionary entry, as in

          CREATE X

       Here the new entry has the name X, but it could have been "Joe"
       or anything else. The new name is a Forth subroutine that will
       return the address of the next available space in memory. Thus

          CREATE X  ok
          X . 247316  ok
          HERE . 247316  ok

       HERE is a subroutine that returns the address of the next available
       space—we note that it is the same as the address of X because no
       space has been ALLOTted. We can rectify this by saying

          10 CELLS ALLOT  ok

       and checking with

          HERE . 247356  ok

       We see that the next available space is now marked as 40 bytes
       further up in memory. (Each CELL is therefore 4 bytes or 32 bits
       on this system.) In other words, the subroutine ALLOT increases
       the pointer HERE by the number of address units you have told
       it to allot. You could have said

         40 ALLOT

       instead of

         10 CELLS ALLOT

       but the latter is more portable because it frees you from having
       to revise your code if you were to run it on a system with 64-bit
       or 16-bit cells (both of which are in common use).


       By executing the sequence

          CREATE X  10 CELLS ALLOT

       we have set aside enough room to hold 10 32-bit numbers--for example
       a table or array--that can be referenced by naming it. If we want to
       get at the 6th element of the array (the first element has index 0,
       so the 6th has index 5) we would say

          X  5 CELLS  +

       to compute its address. To see how this works, let us say

          137  X 5 CELLS +  !  ok

       to store an integer into the 6th array location; then

          X 5 CELLS +  @  . 137  ok

       retrieves and displays it.


       Using the tools provided by CREATE and ALLOT we can devise
       any sort of data structure we like. This is why Forth does
       not provide a panoply of data structures, such as are to be
       found in languages like C, Pascal or Fortran. It is too easy
       in Forth to custom tailor any sort of data structure one
       wishes. In the section on CREATE...DOES> below you will see
       that Forth makes it easy to write subroutines ("constructors")
       that create custom data structures--that can even include
       code fragments that do useful things. For example, a CONSTANT
       is a number you would not want to change during a program's
       execution. So you do not want access to its memory location.
       How then do you get the number when you need it? You package
       the code for @ with the storage location, so that by naming
       the CONSTANT you retrieve its contents. Its usage is

          17 CONSTANT seventeen  ok
          seventeen . 17  ok

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7. Comparing and branching 

       Forth lets you compare two numbers on the stack, using relational 
       operators ">", "<", "=" . Thus, e.g., the phrase 

                2 3 > <cr> ok 

       leaves 0 ("false") on the stack, because 2 (NOS) is not greater than 3 
       (TOS). Conversely, the phrase

                2 3 < <cr> ok 

       leaves -1 ("true") because 2 is less than 3. 

       Notes: In some Forths “true” is +1 rather than -1. 

              Relational operators consume both arguments and leave a “flag” 
              to show what happened. 

       (Many Forths offer unary relational operators "0=", "0>" and "0<". 
       These, as might be guessed, determine whether the TOS contains an 
       integer that is 0, positive or negative.) 

       The relational words are used for branching and control. For example,

           : TEST     0 =  INVERT  IF   CR   ." Not zero!"   THEN  ; 

           0 TEST <cr>  ok     ( no action) 
           -14 TEST <cr> 
           Not zero!  ok 

       The TOS is compared with zero, and the INVERT operator (bitwise logical
       NOT—this flips "true" and "false") is applied to the resulting flag. The
       word CR issues a carriage return (newline). Finally, if TOS is non-zero,
       IF swallows the flag and executes all the words between itself and the 
       terminating THEN. If TOS is zero, execution jumps to the word following
       THEN. 

       The word ELSE is used in the IF...ELSE...THEN statement: a nonzero 
       value in TOS causes any words between IF and ELSE to be executed, and 
       words between ELSE and THEN to be skipped. A zero value produces the 
       opposite behavior. Thus, e.g. 


           : TRUTH    CR   0 =  IF  ." false"  ELSE  ." true"  THEN  ; 

           1 TRUTH <cr> 
           true  ok 

           0 TRUTH <cr> 
           false  ok 

       Since THEN is used to terminate an IF statement rather than in its 
       usual sense, some Forth writers prefer the name ENDIF.

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8. Documenting and commenting Forth code 

       Forth is sometimes accused of being a "write-only" language, i.e. some 
       complain that Forth is cryptic. This is really a complaint against
       poor documentation and untelegraphic word names. Unreadability is 
       equally a flaw of poorly written FORTRAN, PASCAL, C, etc. 

       Forth offers programmers who take the trouble tools for producing ex- 
       ceptionally clear code. 


       a. Parenthesized remarks 

       The word (![](red_box.gif) — a left parenthesis followed by a space — says "disre- 
       gard all following text until the next right parenthesis in the 
       input stream". Thus we can intersperse explanatory remarks within 
       colon definitions. 


       b. Stack comments 

       A particular form of parenthesized remark describes the effect of a
       word on the stack. In the example of a recursive loop (GCD below), 
       stack comments are really all the documentation necessary. 

       Glossaries generally explain the action of a word with a 
       stack-effect comment. For example, 

           ( adr -- n)

       describes the word @ ("fetch"): it says @ expects to find an address 
       (adr) on the stack, and to leave its contents (n) upon completion.  
       The corresponding comment for ! would be 

           ( n adr -- ) . 



       c. Drop line (\) 

       The word "\" (back-slash followed by space) has recently gained favor 
       as a method for including longer comments. It simply means "drop ev- 
       erything in the input stream until the next carriage return". Instruc- 
       tions to the user, clarifications or usage examples are most naturally
       expressed in a block of text with each line set off by "\"  . 



       d. Comment blocks
       ANS Forth contains interpreted IF...THEN, in the form of [IF] ... [THEN].
       Although they are generally used for conditional compilation, these words
       can be used to create comment blocks. Thus we can say

            FALSE [IF]   anything you want to say
            [THEN]

       and the included remarks, code, examples or whatever will be ignored
       by the compiling mechanism.



       e. Self-documenting code 

       By eliminating ungrammatical phrases like CALL or GOSUB, Forth pre-
       sents the opportunity —via telegraphic names for words— to make code
       almost as self-documenting and transparent as a readable English or
       German sentence. Thus, for example, a robot control program could con-
       tain a phrase like

           2 TIMES   LEFT EYE  WINK 

       which is clear (although it sounds like a stage direction for Brun- 
       hilde to vamp Siegfried). It would even be possible without much dif- 
       ficulty to define the words in the program so that the sequence could 
       be made English-like:  WINK  LEFT EYE 2 TIMES . 

       One key to doing this is to eliminate “noise” words like
       @, !, >R, etc. by factoring them out into expressively
       named —and reuseable— subroutines.

       Another is to organize the listing of a subroutine so
       that it physically resembles what it is supposed to do.
       Two examples are the jump table defined below, as well as
       a method for programming finite state automata.

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9. Integer arithmetic operations

       The 1979 or 1983 standards require that a conforming Forth system con- 
       tain a certain minimum set of pre-defined words. These consist of
       arithmetic operators + - * / MOD /MOD */ for (usually) 16-bit signed- 
       integer (-32767 to +32767) arithmetic, and equivalents for unsigned (0
       to 65535), double-length and mixed-mode (16- mixed with 32-bit) arith- 
       metic. The list will be found in the glossary accompanying your 
       system, as well as in SF and FTR. 

       Try this example of a non-trivial program that uses arithmetic and 
       branching to compute the greatest common divisor of two integers using 
       Euclid's algorithm: 

           : TUCK   ( a b -- b a b)   SWAP  OVER  ; 
           : GCD    ( a b -- gcd)  ?DUP  IF  TUCK  MOD  GCD  THEN  ; 

       The word ?DUP duplicates TOS if it is not zero, and leaves it alone 
       otherwise. If the TOS is 0, therefore, GCD consumes it and does 
       nothing else. However, if TOS is unequal to 0, then GCD TUCKs TOS 
       under NOS (to save it); then divides NOS by TOS, keeping the remainder 
       (MOD). There are now two numbers left on the stack, so we again take 
       the GCD of them. That is, GCD calls itself.

       If you try the above code it will fail. A dictionary entry
       cannot be looked up and found until the terminating ";"
       has completed it. So in fact we must use the word RECURSE
       to achieve self-reference, as in
 

           : TUCK   ( a b -- b a b)   SWAP  OVER  ;
           : GCD    ( a b -- gcd)  ?DUP  IF   TUCK  MOD  RECURSE   THEN  ;

       Now try 

           784 48 GCD .  <cr>  16 ok 


       The ANSI/ISO Forth Standard (adopted in 1994) mandates the minimal set
       of arithmetic operators +  -  *  /  MOD  */  /MOD  */MOD and  M* . The
       standard memory-word size is the cell, which must be at least 16 bits,
       but in many modern systems is 32- or even 64 bits wide. Single-length
       integers in Win32Forth are 32 bits. The stack on ANS-compliant Forths
       is always 1 cell wide.

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10. Looping and structured programming

       Forth has several ways to loop, including the implicit method of re- 
       cursion, illustrated above. Recursion has a bad name as a looping
       method because in most languages that permit recursion, it imposes 
       unacceptable running time overhead on the program. Worse, recursion 
       can —for reasons beyond the scope of this Introduction to Forth— be 
       an extremely inefficient method of expressing the problem. In Forth, 
       there is virtually no excess overhead in recursive calls because Forth 
       uses the stack directly. So there is no reason not to recurse if that 
       is the best way to program the algorithm. But for those times when 
       recursion simply isn't enough, here are some more standard methods.

       a. Indefinite loops

       The construct 

           BEGIN xxx ( -- flag)  UNTIL 

       executes the words represented by xxx, leaving TOS (flag) set to TRUE
       —at which point UNTIL terminates the loop— or to FALSE —at which
       point UNTIL jumps back to BEGIN. Try: 

           : COUNTDOWN    ( n --) 
                BEGIN  CR   DUP  .  1 -   DUP   0  =   UNTIL  DROP  ;

           5 COUNTDOWN 
           5 
           4
           3 
           2 
           1  ok 

       A variant of BEGIN...UNTIL is 

           BEGIN xxx ( -- flag) WHILE  yyy  REPEAT

       Here xxx is executed, WHILE tests the flag and if it is FALSE
       leaves the loop; if the flag is TRUE, yyy is executed; REPEAT then
       branches back to BEGIN.

       These forms can be used to set up loops that repeat until some
       external event (pressing a key at the keyboard, e.g.) sets the
       flag to exit the loop. They can also used to make endless loops
       (like the outer interpreter of Forth) by forcing the flag
       to be FALSE in a definition like


           : ENDLESS     BEGIN  xxx  FALSE  UNTIL ;


       b. Definite loops 

       Most Forths allow indexed loops using DO...LOOP (or step +LOOP).
       These are permitted only within definitions 

           : BY-ONES   ( n --)   0 TUCK  DO   CR  DUP  .  1 +  LOOP   DROP  ; 

       The words CR  DUP  .  1 +  will be executed n times as the lower 
       limit, 0, increases in unit steps to n-1. 

       To step by 2's, we use the phrase 2 +LOOP to replace LOOP, as with 

           : BY-TWOS   ( n --)   0 TUCK 
                DO   CR  DUP  .  2 +    2 +LOOP    DROP  ;

       These words can be simplified by accessing the loop index with the word I:

              : BY-TWOS   ( n --)    0  DO   CR   I  .     2 +LOOP  ;

       We can even count backwards, as in launching a rocket, as in

              : countdown   0 SWAP   DO  CR  I  .   -1  +LOOP  ;  ok

              10 countdown
              10
              9
              8
              7
              6
              5
              4
              3
              2
              1
              0  ok


       One may also nest loops and access the index of the outer loop from the inner loop
       using the word J, as in

              : NESTED    ( n m --)  CR
                     0 DO  DUP  ( n n --) 
                            0 DO  CR  J .  I .  
                            LOOP
                     LOOP 
              DROP  ;

              2 3  NESTED  

              0 0
              0 1
              1 0
              1 1
              2 0
              2 1  ok

       Here is something to beware of: suppose the initial indices for the DO loop are equal:
       that is, something like

              17  17  DO   stuff   LOOP

       then the loop will be executed 232-1 times! As the ANS Standard document says,
       “This is intolerable.” Therefore ANS Forth defines a special word, ?DO,  that will skip
       the loop if the indices are equal, and execute it if they are not. It is up to the
       programmer to make sure that if the initial index exceeds the final one, as in  0 17  DO ,
       the program counts down, assuming that is what was intended:

              0 17  DO  stuff   -1  +LOOP



       c. Structured programming 

       N. Wirth invented the Pascal language in reaction to program flow
       charts resembling a plate of spaghetti. Such flow diagrams were
       often seen in early languages like FORTRAN and assembler. Wirth
       intended to eliminate line labels and direct jumps (GOTOs), thereby
       forcing control flow to be clear and direct.

       The ideal was subroutines or functions that performed a single
       task, with unique entries and exits. Unfortunately, programmers
       insisted on GOTOs, so many Pascals and other modern languages now have
       them. Worse, the ideal of short subroutines that do one thing only is
       unreachable in such languages because the method for calling them and 
       passing arguments imposes a large overhead. Thus execution speed re- 
       quires minimizing calls, which in turn means longer, more complex sub- 
       routines that perform several related tasks. Today structured program- 
       ming seems to mean little more than writing code with nested IFs in- 
       dented by a pretty-printer. 

       Paradoxically, Forth is the only truly structured language in common 
       use, although it was not designed with that as its goal. In Forth word 
       definitions are lists of subroutines. The language contains no GOTO's so 
       it is impossible to write “spaghetti” code. Forth also encourages 
       structure through short definitions. The additional running time
       incurred in breaking a long procedure into many small ones (this is
       called “factoring”) is typically rather small in Forth. Each Forth sub-
       routine (word) has one entry and one exit point, and can be written
       to perform a single job.



       d. “Top-down” design 

       “Top-down” programming is a doctrine that one should design the entire 
       program from the general to the particular: 

         > Make an outline, flow chart or whatever, taking a broad overview
           of the whole problem. 

         > Break the problem into small pieces (decompose it). 

         > Then code the individual components. 

       The natural programming mode in Forth is “bottom-up” rather than “top- 
       down” —the most general word appears last, whereas the definitions 
       must progress from the primitive to the complex. This leads to a some- 
       what different approach from more familiar languages: 

         > In Forth, components are specified roughly, and then as they are 
           coded they are immediately tested, debugged, redesigned and 
           improved. 

         > The evolution of the components guides the evolution of the outer 
           levels of the program.

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11. CREATE ... DOES> (the pearl of FORTH) 

       Michael Ham has called the word pair CREATE...DOES>, the “pearl of 
       Forth”. CREATE is a component of the compiler, whose function is to
       make a new dictionary entry with a given name (the next name in the 
       input stream) and nothing else. DOES> assigns a specific run-time ac- 
       tion to a newly CREATEd word. 


       a. Defining “defining” words 

       CREATE finds its most important use in extending the powerful class of 
       Forth words called “defining” words. The colon compiler  ":"  is such 
       a word, as are VARIABLE and CONSTANT. 

       The definition of VARIABLE in high-level Forth is simple 

           : VARIABLE  CREATE   1 CELLS  ALLOT ;

       We have already seen how VARIABLE is used in a program. (An altern-
       ative definition found in some Forths is

           : VARIABLE  CREATE   0  ,  ;

       —these variables are initialized to 0.)

       Forth lets us define words initialized to contain specific values: for
       example, we might want to define the number 17 to be a word. CREATE
       and "," ("comma") can do this:

           17 CREATE SEVENTEEN  ,  <cr>  ok

       Now test it via

           SEVENTEEN @ .  <cr>  17 ok .


       Remarks: 

         > The word , ("comma") puts TOS into the next cell of the dic-
           tionary and increments the dictionary pointer by that number of
           bytes.

         > A word "C," ("see-comma") exists also — it puts a character into
           the next character-length slot of the dictionary and increments
           the pointer by 1 such slot. (In the ASCII character representation
           the slots are 1 byte long; Unicode characters require 2 bytes.)


       b. Run-time vs. compile-time actions 

       In the preceding example, we were able to initialize the variable 
       SEVENTEEN to 17 when we CREATEd it, but we still have to fetch it to 
       the stack via SEVENTEEN  @ whenever we want it. This is not quite what
       we had in mind. We would like to find 17 in TOS when SEVENTEEN is 
       named. The word DOES> gives us the tool to do this. 

       The function of DOES> is to specify a run-time action for the “child” 
       words of a defining word.  Consider the defining word CONSTANT , de- 
       fined in high-level (of course CONSTANT is usually defined in machine
       code for speed) Forth by

           : CONSTANT  CREATE  ,  DOES>  @  ; 

       and used as 

           53 CONSTANT PRIME  <cr> ok 

       Now test it:

           PRIME . <cr>  53  ok . 


       What is happening here? 

         > CREATE (hidden in CONSTANT) makes an entry named PRIME (the
           first word in the input stream following CONSTANT). Then "," 
           places the TOS (the number 53) in the next cell of the dic-
           tionary.

         > Then DOES> (inside CONSTANT) appends the actions of all words be- 
           tween it and ";" (the end of the definition) —in this case, "@"— 
           to the child word(s) defined by CONSTANT. 


       c. Dimensioned data (intrinsic units) 

       Here is an example of the power of defining words and of the distinc- 
       tion between compile-time and run-time behaviors. 

       Physical problems generally involve quantities that have dimensions,
       usually expressed as mass (M), length (L) and time (T) or products of 
       powers of these. Sometimes there is more than one system of units in
       common use to describe the same phenomena.

       For example, U.S. or English police reporting accidents might use 
       inches, feet and yards; while Continental police would use centimeters 
       and meters. Rather than write different versions of an accident ana- 
       lysis program it is simpler to write one program and make unit conver- 
       sions part of the grammar. This is easy in Forth. 

       The simplest method is to keep all internal lengths in millimeters, 
       say, and convert as follows: 

                : INCHES  254   10  */ ; 
                : FEET   [ 254 12 * ] LITERAL  10  */ ; 
                : YARDS  [ 254 36 * ] LITERAL  10  */ ; 
                : CENTIMETERS   10  * ; 
                : METERS   1000  * ; 

       Note: This example is based on integer arithmetic.  The word */
             means “multiply the third number on the stack by NOS, keeping
             double precision, and divide by TOS”. That is, the stack com-
             ment for */ is ( a b c -- a*b/c). 


       The usage would be 

                10 FEET  .  <cr>  3048 ok


       The word "[" switches from compile mode to interpret mode while com-
       piling. (If the system is interpreting it changes nothing.) The word 
       "]" switches from interpret to compile mode. 

       Barring some error-checking, the “definition” of the colon compiler 
       ":" is just 

           :  :   CREATE  ]  DOES>  doLIST  ;

       and that of ";" is just 

           :  ;   next  [  ;  IMMEDIATE

       Another use for these switches is to perform arithmetic at compile-
       time rather than at run-time, both for program clarity and for easy 
       modification, as we did in the first try at dimensioned data (that is, 
       phrases such as 

           [ 254 12 * ] LITERAL 

       and 

           [ 254 36 * ] LITERAL

       which allowed us to incorporate in a clear manner the number of 
       tenths of millimeters in a foot or a yard.


       The preceding method of dealing with units required unnecessarily many
       definitions and generated unnecessary code. A more compact approach
       uses a defining word, UNITS :

           : D,  ( hi lo --)   SWAP  , ,  ;
           : D@  ( adr -- hi lo)   DUP  @   SWAP   CELL+  @   ;
           : UNITS  CREATE  D,   DOES> D@  */ ;

       Then we could make the table

                254 10        UNITS INCHES
                254 12 *  10  UNITS FEET
                254 36 *  10  UNITS YARDS
                10  1         UNITS CENTIMETERS
                1000  1       UNITS METERS

                \ Usage:
                10 FEET  . <cr>  3048  ok
                3 METERS . <cr>  3000  ok
                \ .......................
                \ etc.

       This is an improvement, but Forth permits a simple extension that
       allows conversion back to the input units, for use in output:

           VARIABLE  <AS>    0 <AS> !
           : AS     TRUE  <AS> ! ;
           : ~AS    FALSE <AS> ! ;
           : UNITS  CREATE  D,  DOES>  D@  <AS> @
                    IF  SWAP  THEN
                    */    ~AS  ;

           \ UNIT DEFINITIONS REMAIN THE SAME.
           \ Usage:
           10 FEET  .   <cr>  3048  ok
           3048 AS FEET  .  <cr>  10  ok



       d. Advanced uses of the compiler

       Suppose we have a series of push-buttons numbered 0-3, and a word WHAT
       to read them. That is, WHAT waits for input from a keypad: when button
       #3 is pushed, for example, WHAT leaves 3 on the stack.

       We would like to define a word BUTTON to perform the action of pushing
       the n'th button, so we could just say:

           WHAT BUTTON

       In a conventional language BUTTON would look something like

           : BUTTON  DUP  0 =  IF  RING  DROP  EXIT  THEN
                     DUP  1 =  IF  OPEN  DROP  EXIT  THEN
                     DUP  2 =  IF  LAUGH DROP  EXIT  THEN
                     DUP  3 =  IF  CRY   DROP  EXIT  THEN
                     ABORT" WRONG BUTTON!"   ;

       That is, we would have to go through two decisions on the average.

       Forth makes possible a much neater algorithm, involving a “jump
       table”. The mechanism by which Forth executes a subroutine is to
       feed its “execution token” (often an address, but not necessarily)
       to the word EXECUTE. If we have a table of execution tokens we need
       only look up the one corresponding to an index (offset into the table)
       fetch it to the stack and say EXECUTE.

       One way to code this is

            CREATE  BUTTONS  ' RING ,  ' OPEN ,  ' LAUGH ,  ' CRY ,
            : BUTTON   ( nth --)    0 MAX  3 MIN
                    CELLS  BUTTONS  +  @  EXECUTE  ;

       Note how the phrase 0 MAX  3 MIN protects against an out-of-range 
       index. Although the Forth philosophy is not to slow the code with un-
       necessary error checking (because words are checked as they are de- 
       fined), when programming a user interface some form of error handling
       is vital. It is usually easier to prevent errors as we just did, than 
       to provide for recovery after they are made. 

       How does the action-table method work? 

         > CREATE BUTTONS makes a dictionary entry BUTTONS.

         > The word '  (“tick”) finds the execution token (xt) of the
           following word, and the word , (“comma”) stores it in the
           data field of the new word BUTTONS. This is repeated until
           all the subroutines we want to select among have their xt's
           stored in the table.

         > The table BUTTONS now contains xt's of the various actions of
           BUTTON.

         > CELLS then multiplies the index by the appropriate number of
           bytes per cell, to get the offset into the table BUTTONS
           of the desired xt.

         > BUTTONS +  then adds the base address of BUTTONS to get the abso-
           lute address where the xt is stored.

         > @ fetches the xt for EXECUTE to execute.

         > EXECUTE then executes the word corresponding to the button pushed.
           Simple!

       If a program needs but one action table the preceding method suffices.
       However, more complex programs may require many such. In that case
       it may pay to set up a system for defining action tables, including
       both error-preventing code and the code that executes the proper
       choice. One way to code this is

            : ;CASE   ;                     \ do-nothing word

            : CASE:
                CREATE  HERE  -1  >R   0  ,   \ place for length
                BEGIN   BL  WORD  FIND        \ get next subroutine
                   0=  IF   CR  COUNT  TYPE  ."  not found"  ABORT  THEN
                   R>  1+  >R
                   DUP  ,    ['] ;CASE  =
                UNTIL   R>   1-  SWAP  !      \ store length
                DOES>   DUP  @   ROT          ( -- base_adr len n)
                        MIN  0  MAX           \ truncate index
                        CELLS  +  CELL+  @  EXECUTE  ;

       Note the two forms of error checking. At compile-time, CASE:
       aborts compilation of the new word if we ask it to point to an
       undefined subroutine:

            case: test1   DUP  SWAP  X  ;case
            X not found

       and we count how many subroutines are in the table (including
       the do-nothing one, ;case) so that we can force the index to
       lie in the range [0,n].

            CASE:  TEST  *  /  +  -  ;CASE  ok
            15 3 0 TEST . 45  ok
            15 3 1 TEST . 5  ok
            15 3 2 TEST . 18  ok
            15 3 3 TEST . 12  ok
            15 3 4 TEST . . 3 15  ok

       Just for a change of pace, here is another way to do it:

          : jtab:  ( Nmax --)      \ starts compilation
               CREATE              \ make a new dictionary entry
               1-  ,               \ store Nmax-1 in its body
          ;                        \ for bounds clipping

          : get_xt    ( n base_adr -- xt_addr)
               DUP  @      ( -- n base_adr Nmax-1)
               ROT         ( -- base_adr Nmax-1 n)
               MIN  0  MAX    \ bounds-clip for safety
               1+  CELLS+  ( -- xt_addr = base + 1_cell + offset)
          ;

          : |   '  ,   ;     \ get an xt and store it in next cell

          : ;jtab   DOES>  ( n base_adr --)   \ ends compilation
                    get_xt  @  EXECUTE        \ get token and execute it
          ;    \ appends table lookup & execute code

          \ Example:
          : Snickers   ." It's a Snickers Bar!"   ;   \ stub for test

          \ more stubs

          5 jtab:  CandyMachine
                   | Snickers
                   | Payday
                   | M&Ms
                   | Hershey
                   | AlmondJoy
          ;jtab

          3 CandyMachine  It's a Hershey Bar!   ok
          1 CandyMachine  It's a Payday!   ok
          7 CandyMachine  It's an Almond Joy!   ok
          0 CandyMachine  It's a Snickers Bar!   ok
         -1 CandyMachine  It's a Snickers Bar!   ok

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12. Floating point arithmetic

       Although Forth at one time eschewed floating point arithmetic
       (because in the era before math co-processors integer arithmetic
       was 3x faster), in recent years a standard set of word names has
       been agreed upon. This permits the exchange of programs that will
       operate correctly on any computer, as well as the development of
       a Scientific Subroutine Library in Forth (FSL).

       Although the ANS Standard does not require a separate stack for
       floating point numbers, most programmers who use Forth for numer-
       ical analysis employ a separate floating point stack; and most of
       the routines in the FSL assume such. We shall do so here as well.

       The floating point operators have the following names and perform
       the actions indicated in the accompanying stack comments:

            F@      ( adr --)       ( f: -- x)
            F!      ( adr --)       ( f: x --)
            F+                      ( f: x y -- x+y)
            F-                      ( f: x y -- x-y)
            F*                      ( f: x y -- x*y)
            F/                      ( f: x y -- x/y)
            FEXP                    ( f: x -- e^x)
            FLN                     ( f: x -- ln[x])
            FSQRT                   ( f: x -- x^0.5)

       Additional operators, functions, trigonometric functions, etc. can
       be found in the FLOATING and FLOATING EXT wordsets. (See dpANS6—
       available in HTML, PostScript and MS Word formats. The HTML version
       can be accessed from this homepage.)

       To aid in using floating point arithmetic I have created a simple
       FORTRAN-like interface for incorporating formulas into Forth words.

       The file ftest.f (included below) illustrates how ftran201.f
       should be used.

\ Test for ANS FORmula TRANslator

marker -test
fvariable a
fvariable b
fvariable c
fvariable d
fvariable x
fvariable w

: test0   f" b+c"  cr  fe.
          f" b-c"  cr  fe.
          f" (b-c)/(b+c)"  cr fe.  ;

3.e0 b f!
4.e0 c f!
see test0
test0

: test1   f" a=b*c-3.17e-5/tanh(w)+abs(x)"  a f@  cr fe.  ;
1.e-3 w f!
-2.5e0 x f!
cr cr
see test1
test1

cr cr
: test2   f" c^3.75"  cr fe.
          f" b^4"     cr fe.  ;
see test2
test2

\ Baden's test case

: quadroot c f! b f! a f!
      f" d = sqrt(b^2-4*a*c) "
      f" (-b+d)/(2*a) "  f" (-b-d)/(2*a) "
;
cr cr
see quadroot

: goldenratio  f" max(quad root(1,-1,-1)) "  ;
cr cr
see goldenratio
cr cr
goldenratio f.



0 [IF]
Output should look like:

: test0
  c f@ b f@ f+ cr fe. c f@ fnegate b f@ f+ cr fe. c f@ fnegate b f@
  f+ c f@ b f@ f+ f/ cr fe. ;
7.00000000000000E0
-1.00000000000000E0
-142.857142857143E-3


: test1
  x f@ fabs 3.17000000000000E-5 w f@ ftanh f/ fnegate b f@ c f@ f* f+
  f+ a f! a f@ cr fe. ;
14.4682999894333E0  ok

: test2
  c f@ noop 3.75000000000000E0 f** cr fe. b f@ f^4 cr fe. ;
181.019335983756E0
81.0000000000000E0  ok

: QUADROOT      C F! B F! A F! B F@ F^2 flit 4.00000 A F@
                C F@ F* F* F- FSQRT D F! B F@ FNEGATE D
                F@ F+ flit 2.00000 A F@ F* F/ B F@ FNEGATE
                D F@ F- flit 2.00000 A F@ F* F/ ;


: GOLDENRATIO           flit 1.00000 flit -1.00000 flit -1.00000
                QUADROOT FMAX ;

1.61803  ok

with more or fewer places.

[THEN]

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13. Non-trivial programming example

       To illustrate how to construct a non-trivial program, let
       us develop a binary search root-finder. We will use the
       FORmula TRANslator ftran201.f to simplify the appearance
       of the code (that is, it hides the data fetches and
       stores that would otherwise be required).

       First we need to understand the algorithm thoroughly:

       If we know that the roots are bracketed between xa and
       xb, and that f(xa)*f(xb) < 0  (at least 1 root lies in
       the interval) we take the next guess to be xp = (xa+xb)/2 .

       We then evaluate the function at xp: fp = f(xp).
       If fa*fp > 0 we set xa = xp, else we set xb = xp.
       We repeat until the ends of the interval containing
       the root are sufficiently close together.


       To begin programming, we note that we will have to keep
       track of three points: xa, xb and xp. We also have to
       keep track of three function values evaluated at those
       points, Ra, Rb and Rp. We also need to specify a pre-
       cision, epsilon, within which we expect to determine
       the root.

       Next we need to define the user interface. That is, once
       we have a subroutine that finds roots, how will we invoke
       it? Since we would like to be able to specify the name of
       the function to find the root of at the same time we
       specify the interval we think the root is in, we need
       some way to pass the name to the root finder as an
       argument.

       I have previously developed an interface that suits me: I
       say

            use( fn.name xa xb precision )bin_root

       as in

            use( f1  0e0 2e0 1e-5 )bin_root

       and the root will be left on the floating point stack.


       The code for passing names of functions as arguments is
       included when you load ftran201.f — the words used in
       this program are use( , v: and defines .  v: creates a
       dummy dictionary entry (named dummy in the program)
       which can be made to execute the actual function whose
       name is passed to the word )bin_root .

       Here are the data structures and their identifications:

        MARKER -binroots    \ say -binroots to unload

        \ Data structures

            FVARIABLE Ra                      \ f(xa)
            FVARIABLE Rb                      \ f(xb)
            FVARIABLE Rp                      \ f(xp)
            FVARIABLE xa                      \ lower end
            FVARIABLE xb                      \ upper end
            FVARIABLE xp                      \ new guess
            FVARIABLE epsilon                 \ precision

            v: dummy                          \ create dummy dictionary entry

       The actual root-finding subroutine, )bin_root , will be
       quite simple and easy to follow:

               : )bin_root  ( xt --)   ( f: Low High Precision -- root)
                   initialize
                   BEGIN   NotConverged?   WHILE   NewPoint   REPEAT
                   f" (xa+xb)/2"           ( f: -- root)
               ;

       Note that the subroutines comprising it are telegraph-
       ically named so they need no explanation; whereas
       )bin_root itself is explained by its stack comments. The
       comments on the first line indicate that )bin_root expects
       an “execution token” on the data stack, and three floating
       point numbers on the floating point stack. These are its
       arguments. (See 11d for a discussion of EXECUTE, etc.)
       The execution token is what is used to change the
       behavior of the dummy dictionary entry dummy : we say

            defines dummy

       in the word initialize to make dummy behave like the
       function whose root we are seeking.

       The final comment ( f: -- root) indicates that )bin_root leaves
       the answer on the floating point stack.


       In a sense we are programming from the top down, since we
       have begun with the last definition of the program and
       are working our way forward. In Forth we often go both
       ways —top-down and bottom-up— at the same time.


       The key words we must now define are initialize ,
       NotConverged? and NewPoint . We might as well begin with
       initialize since it is conceptually simple:

               : initialize    ( xt --) ( f: lower upper precision --)
                   defines dummy                       \ xt -> DUMMY
                   f" epsilon="    f" xb="   f" xa="   \ store parameters
                   f" Ra=dummy(xa)"
                   f" Rb=dummy(xb)"
                   f" MoreThan( Ra*Rb, 0)"             \ same sign?
                   ABORT" Even # of roots in interval!"
               ;

       The word ABORT" prints the message that follows it and
       aborts execution, if it encounters a TRUE flag on the
       data stack. It is widely used as a simple error handler.
       ABORT (without the " ) simply aborts execution when
       it is encountered. So it usually is found inside some
       decision structure like an IF...THEN clause. (See 11d for
       two examples of usage.)

       ABORT" was preceded by a test. In order to use a test as
       a function in a Fortran-like expression (this test con-
       sumes two arguments from the floating point stack and
       leaves a flag on the data stack), we must define a synonym
       for it. The reason is that ftran201.f does not recognize
       relational operators like > or < . The definition is*

               : MoreThan    ( f: a b)  ( -- true if a>b)
                   POSTPONE  F>  ;  IMMEDIATE

       The code produced by f" MoreThan( Ra*Rb, 0)" is then just

               RA F@ RB F@ F* flit 0.00000E-1 F>

       which is what we want. We have already explained the
       phrase defines dummy. The phrases f" xa=" and so on are
       shorthand for storing something from the floating point
       stack to a floating point variable. Thus f" xa="
       generates the code XA F! . The rest of initialize is to
       calculate the function at the endpoints of the supposed
       bounding interval (a,b).


       NotConverged? is a test for (non)convergence. WHILE
       expects a flag on the data stack, as described in 10a. So
       we define

               : NotConverged?    ( -- f)
                   f" MoreThan( ABS( xa - xb ), epsilon )"   ;

       which generates the code

               XB F@ XA F@ F- FABS EPSILON F@ F>


       What about NewPoint ? Clearly,

               : NewPoint
                   f" xp = (xa+xb)/2"      \ new point
                   f" Rp = dummy(xp)"
                   f" MoreThan( Ra*Rp, 0)" \ xp on same side of root as xa?

                   IF      f" xa=xp"  f" Ra=Rp"
                   ELSE    f" xb=xp"  f" Rb=Rp"   THEN
               ;

       That is, we generate a new guess by bisection, evaluate the
       function there and decide how to choose the new bounding
       interval.

       All that remains is to put the definitions in the proper order
       and test the result by loading the program bin_rts.f and
       trying out the test case.


        FALSE [IF]
        Usage example:

            : f1   fdup  fexp  f*  1e0  f-  ;  ok
            use( f1  0e0 2e0 1e-5 )bin_root f. .567142  ok

        [THEN]

       Finally, if we want to be very careful indeed, and/or are
       planning to re-use the program, we add an appropriate
       boilerplate header, such as that included in the file
       bin_rts.f .



---


      *Note: the word POSTPONE in this context means that the word following it
       —in this case F> — will be compiled into the word that uses MoreThan rather
       than in MoreThan itelf. (Note that MoreThan is IMMEDIATE.) This way of doing
       things saves some overhead during execution. Some Forths (notably Win32Forth)
       define a word SYNONYM to accomplish the same thing.


---



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14. Some remarks about strings
       As in other languages, alphanumeric strings in Forth are represented as
       contiguous arrays in memory, each memory unit being a “character”.
       Traditionally a character encoded by the ASCII or EBCDIC systems occupied
       one (1) byte of storage, allowing for 256 characters. With the need to encode
       alphabets other than the Latin one (e.g. Chinese, Arabic, Hebrew, Cyrillic) a
       two-byte encoding called Unicode has been adopted, which allows for 65535
       distinct characters.

       A traditional Forth string consisted of a count byte and up to 255 bytes containing
       alphanumeric characters (usually in ASCII). In ANS Forth this scheme has been
       abandoned: how strings are stored will depend on the implementation. However
       ANS Forth contains words that enable us to manipulate strings without reference
       to how they are implemented.

       Most ANS Forths (and Win32Forth is one of them) define S" to have defined
       interpretive as well as compiling behavior. This means that if we say

              S" This is a string!" CR TYPE

       we get

              This is a string! ok

       What happened? S" This is a string!" created a string with text beginning at a
       “c-address” and with a “count” that says how many characters (including blanks)
       the string includes. The address and count are left on the stack. That is, the
       proper stack picture would be

              S" This is a string!"   ( -- c-addr u)

       (the count is an unsigned integer u because strings of negative length are
       meaningless).

       The word CR means “insert a carriage return”, and TYPE means “from the 
       c-addr output u characters to the screen”.

       Exercise:
       Use what you have just learned to write a “Hello world!” program.


       It is perfectly feasible to define one's own word set for working with strings, depending
       on what sort of application one is writing. For example, I have written a program to
       translate mathematical formulas in Fortran-like form into Forth code, outputting the
       result either to the screen (for test purposes) or embedding it into a Forth definition.
       There is even a variant that evaluates the formula, provided all the variables have
       been previously defined and given numerical values. To accomplish this required
       strings longer than 255 characters, so I defined my own.



       I now want to turn to “pictured numerical output”. Many computer programs
       need to output numbers in some particular format, no matter how they are stored intern-
       ally. For example an accounting program might output monetary amounts in the usual
       dollars-and-cents format. The Forth words that accomplish this are

              # ,  <# ,  #S ,  #> , SIGN and HOLD

       They do not have any defined interpretive behavior (although there is no telling what
       any particular Forth may do) and are intended to be used within word definitions. Here
       is an example: suppose we are writing an accounting program. Since most users will
       not be dealing with amounts that exceed $100,000,000 we can use signed 32-bit integers
       to represent the dollars and cents. (Such numbers can represent amounts up to
       ±(231—1) = ±2147483647 cents.) Signed double-length integers are at least 32 bits long
       on all ANS-compatible systems (although they will be 64 bits on 32-bit computers).
       Hence we shall use doubles so the program will run on any ANS-compatible
       Forth.

       A double-length integer is entered from the keyboard by including a decimal point in
       it, as

              -4756.325  ok

       Let us define a word to output a double-length integer. The first part will be to
       translate it to an alphanumeric string referred to by c-addr u.

          : (d.$)    ( d -- c-adr u)  TUCK DABS  <#  # # [CHAR] .  HOLD  #S  ROT  SIGN  #>  ;

       As the stack comment ( d -- c-adr u) shows, (d.$) consumes a (signed) double-length
       integer from the stack and leaves the string data in a form that can be printed to the
       screen by the word TYPE. Let us test this:

              4376.58  (d.$)  CR  TYPE
              4376.58 ok

              -4376.99  (d.$)  CR  TYPE
              -4376.99 ok

       It is worth exploring what each part does. A double length integer is stored as two
       cells on the stack, with the most-significant part on TOS. Thus the word TUCK
       places the most-significant part (containing the algebraic sign) above d and then DABS
       converts d to |d|. Next, <# begins the process of constructing an alphanumeric 
       string. The two instances of # peel off the two least-significant digits and put
       them in the string. The phrase [CHAR] .  HOLD adds a decimal point to the string.

       [CHAR] builds in the representation of the character . as a numeric literal (in
       ASCII it is 46). HOLD then adds it to the string under construction. (HOLD has no
       meaning except between <# and #>.) Then the word #S takes the rest of the digits
       and adds them to the nascent string.

       (Semi)finally, ROT puts the most significant part of d (with its sign) on TOS, and
       SIGN adds its algebraic sign to the beginning of the string. (Again, SIGN is only
       meaningful between <# and #>.) 

       And finally, the word #> cleans everything up and leaves c-addr u
       on the stack, ready for display or whatever.

       Exercises:

       a) How would you add a leading dollar sign ($) to the output number?
       b) How would you enclose a negative amount in parentheses rather than
          displaying a — sign? [That is, ( 4376.99) rather than -4376.99.]
       c) Define a word to display a double-length integer in dollar-and-cents format.

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15. Assembly language programming
       Most Forths include an assembler that makes programming in machine code almost as
       easy as programming in high level Forth. Why would one want to do that? There are
       reallly only two reasons for dropping into machine language:

       > One must perform a task requiring carnal knowledge of the hardware;
       > Part of the program must be optimized for speed.

       In this section we deal only with the second reason. We imagine that after careful
       algorithmic analysis there is no way to further speed up a high level program. However
       the requirements of the application demand a substantial speed improvement. Since
       most Forths are somewhat inefficient relative to optimized C or Fortran, there may be
       a substantial speed gain to be realized from hand-coding in assembly language. An
       example is the innermost loop in a linear equations solver. For n equations it is
       executed n3 times. Moreover it is a very simple loop, containing 2 fetches, a multi-
       plication, a subtraction and a store. Thus it is a perfect candidate to be optimized.
       By contrast, the middle- and outermost loops are executed respectively n2 and n times,
       so there is little point in optimizing them (that is, for small matrices the running
       time is too short to care; whereas for large matrices— n > 100 —the middle loop would
       have to run 100× faster to be worth rewriting in machine code.

       It is important to realize that assembly language conventions differ from
       Forth to Forth. Moreover the instruction set will be particular to a given
       target computer. That is, there is no such thing as a generic assembler
       in any programming environment, much less for Forth. Hence everything
       we do here will be specific to Win32Forth running on a Pentium-class machine.

       We begin with a little warmup exercise. Suppose I found that my program
       used the sequence *  +  many times. Obviously good factoring practice
       would dictate that this sequence be given its own name and defined as
       a new subroutine (word). So we might define

              : *+   *  +  ;

       and substitute it for the sequence *  +  throughout the program. But
       suppose we discover that this short sequence is the bottleneck in our
       program's running time, so that speeding it up will greatly increase
       speed of execution. (I realize it isn't likely for this example---bear
       with me!) So we would like to translate it into machine code. To do
       this we first look at the machine code for * and + separately. These
       are primitive words and almost certainly will be CODE definitions in
       any reasonable Forth.

       Thus we need to disassemble these words. In some Forths this might mean
       inspecting the contents of the word byte by byte, and looking up the code
       sequences in the operating manual for that cpu. Fortunately for us,
       Win32Forth has a built-in disassembler. If we SEE a CODEd definition it
       will return the actual byte-codes as well as the names of the instructions
       in the Win32Forth assembler. Let us try this out: we get

              see +
              + IS CODE
                      4017AC 58               pop     eax
                      4017AD 03D8             add     ebx, eax

                       ok

              see *
              * IS CODE
                      401B9C 8BCA             mov     ecx, edx
                      401B9E 58               pop     eax
                      401B9F F7E3             mul     ebx
                      401BA1 8BD8             mov     ebx, eax
                      401BA3 8BD1             mov     edx, ecx

                       ok


       To understand these sequences we must bear in mind that Win32Forth keeps
       TOS in a 32-bit register, in fact the ebx register. We must also know that
       Win32Forth uses the edx register for something or other—probably to do
       with the mechanism for executing a word and returning control to the next
       word in the program (that is, the threading mechanism). So if a program
       is going to modify the edx register, its previous contents have to be
       saved somewhere. Since addition of eax to ebx does not affect edx, the
       CODE for + doesn't need to protect edx; however, when two 32-bit numbers
       are multiplied, the result can contain as many as 64 bits. Thus the product
       occupies the two registers eax (bits 0 through 31) and edx (bits 32-63).

       This is the reason for saving edx into the unused ecx register, and then
       restoring it afterward.

       It is worth noting, before we go too far, that the Win32Forth assembler
       preserves the Intel conventions. That is,

              add     ebx, eax

       adds the contents of register eax to ebx, leaving the result in ebx (which
       is where we want it because that is TOS). Similarly, the sequences

              mov     ecx, edx

       and

              mov     ebx, eax

       have the structure

              mov     destination, source

       We should also ask why the integer multiplication instruction

              mul     ebx

       has only one operand. The answer is that the register eax is the so-called
       “accumulator”, so it contains one of the multiplicands initially and then it
       and edx contain the product, as noted above. It is then only necessary to
       specify where the other multiplicand is coming from (it could be a cell in
       memory).

       Therefore to define the word *+ in assembler we would type in

              CODE *+             ( a b c -- b*c+a) \ stack: before -- after
                  mov ecx, edx    \ protect edx because mul alters it
                  pop eax         \ get item b; item c (TOS) is already in ebx
                  mul ebx         \ integer multiply-- c*b -> eax (accumulator)
                  pop ebx         \ get item a
                  add ebx, eax    \ add c*b to a -- result in ebx (TOS) --done
                  mov edx, ecx    \ restore edx
                  next,           \ terminating code for Forth interpreter
              END-CODE  ok

       Note that the Forth assembler recognizes Forth comments—Intel-style
       comments would be preceded by semicolons, but we obviously can't use these
       because semicolon is a Forth word.

       The word END-CODE has an obvious meaning, but what about next, (the comma
       is part of the name and is significant!). Advanced users of the assembler
       sometimes need to define code sequences that do not include the instructions
       to transfer control to the next word. So Win32Forth has factored this function
       out of the CODE terminating sequence. For this example we require these
       instructions to be assembled, so we include next,  .


       Before going further, you should try out this example and convince yourself
       it works.


       For our nontrivial example we are going to hand-code the innermost loop of
       my linear equations solver. I programmed this in high-level Forth in the
       form

        : }}r1-r2*x   ( M{{ r1 r2 -- )  ( f: x -- x)  \ initialize assumed
            0 0
            LOCALS| I1 I2 r2  r1  mat{{ |           \ local names
            frame| aa |                             \ local fvariable
            Iperm{ r1 } @  TO  I1
            Iperm{ r2 } @  TO  I2
            Nmax  r2  ?DO                           \ begin loop
                f" mat{{ I1 I }} = mat{{ I1 I }}
                   - mat{{ I2 I }} * aa"
            LOOP                                    \ end loop
            f" aa"  |frame                          ( f: -- x)
        ;

       Here Iperm{ is the name of an array of integers that holds the permuted
       row-labels; note that the rows we work on do not change within the actual
       loop. Neither does the floating point number represented by the local
       variable aa. What does change are the row-elements. 


       To translate }}r1-r2*x to assembler we will need to factor it a bit more
       finely. Evidently we are subtracting the I'th element of row I2, multi-
       plied by aa, from the I'th element of row I1. Moreover, since the matrix
       has been partially triangularized already, we do not start with element 0
       but with element r2. Finally, as we have noted previously, ?DO includes
       a bounds check so that if r2 equals or is greater than Nmax the loop is
       not executed. So we shall revise }}r1-r2*x to include this test explicitly
       and CODE only the loop itself. That is, we shall write


        : incr_addrs   ( addr1  addr2 -- addr1+inc addr2+inc)
              [ 1 FLOATS ]  LITERAL
              TUCK  +   -ROT  +  SWAP  ;

        : inner_loop  ( addr1 addr2 Nmax lower_limit -- ) ( f: x -- x)
              DO                                 \ begin loop
                 ( f: aa)  ( addr1 addr2)        \ loop invariant
                 2DUP   SWAP  F@                 ( f: aa m[I1,I])
                 FOVER  F@ F*   F-  OVER  F!
                     \  m[I1,I = m[I1,I] - m[I2,I]*x
                 ( f: x)  ( addr1 addr2)        \ loop invariant
                 incr_addrs                      \ increment addresses
              LOOP                               \ end loop
              2DROP
        ;

        : }}r1-r2*x   ( M{{ r1 r2 -- )  ( f: x -- x)  \ initialize assumed
            0 0
            LOCALS| I1 I2 r2  r1  mat{{ |           \ local names
            Iperm{ r1 } @  TO  I1
            Iperm{ r2 } @  TO  I2
            Nmax  r2  >
            IF   mat{{ I1 r2 }}  mat{{ I1 r2 }}     \ base adresses
                 Nmax  r2                                \ loop limits
                 ( f: x)  ( addr1 addr2 Nmax lower_limit)
                       inner_loop
                 ( f: x)  ( -- )
            THEN
        ;

       So what we are going to CODE here is the word inner_loop, since these are
       the only instructions executed n3 times.

       CODE inner_loop      ( addr1 addr2 Nmax lower_limit -- )  ( f: x -- x)
          fld FSIZE FSTACK_MEMORY                \ f: -> fpu:
          mov ecx, ebx                           \ ecx = r2
          pop eax                                \ eax = Nmax
                                                 ( addr1 ebx=addr2)
          push edx                               ( addr1 edx ebx)
          mov edx, 4 [esp]                       \ edx = addr1)
                                          \ begin loop
       L$1:  fld [ebx] [edi]                     ( fpu: aa m[addr2]
          fmul st, st(1)                         ( fpu: aa m2*aa)
          fld [edx] [edi]                        ( fpu: aa m2*aa m1)
          fxch st(1)                             ( fpu: aa m1 m2*aa)
          fsubp st(1), st                        ( fpu: aa m1-m2*aa)
          fstp  [edx] [edi]                      ( fpu: aa)
          add [edx], # 8                         \ increment addresses
          add [ebx], # 8
          inc ecx                                \ add 1 to loop variable
          cmp eax, ecx                           \ test for done
          jl  L$1                                \ loop if I < Nmax
                                          \ end loop
          pop edx                                \ restore edx
          pop ebx
          pop ebx                                \ clean up data stack
          fstp FSIZE FSTACK_MEMORY               \ fpu: -> f:
          next,
       END-CODE

       A final remark: I have written a tool for translating automatically a
       sequence of floating point operations to CODE for the Intel fpu. This
       tool, ctran.f, is specialized for Win32Forth.

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16. Some useful references
             > M. Kelly and N. Spies, Forth: A text and Reference (Prentice-Hall, NJ, 1986)  

             > L. Brodie, Starting Forth, 2nd ed. (Prentice-Hall, NJ, 1986)  

             > L. Brodie, Thinking FORTH (Prentice-Hall, NJ, 1984 (online edition)  


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