Research V6 development

Work on file usr/doc/bc/bca
Work on file usr/doc/bc/dc
Work on file usr/doc/bc/bc

Co-Authored-By: Robert Morris <rhm@research.uucp>
Synthesized-from: v6

diff --git a/usr/doc/bc/bc b/usr/doc/bc/bc
new file mode 100644 (file)
index 0000000..6ea0cad
--- /dev/null
+++ b/usr/doc/bc/bc
@@ -0,0 +1,614 @@
+.RP
+.TL
+BC \- An Arbitrary Precision Desk-Calculator Language
+.AU
+Lorinda Cherry
+.AU
+Robert Morris
+.AI
+.MH
+.AB
+BC is a language and a compiler for doing arbitrary precision arithmetic
+on the PDP-11 under the UNIX time-sharing
+system.  The output of the compiler is interpreted and executed by
+a collection of routines which can input, output, and do
+arithmetic on indefinitely large integers and on scaled fixed-point
+numbers.
+.PP
+These routines are themselves based on a dynamic storage allocator.
+Overflow does not occur until all available core storage
+is exhausted.
+.PP
+The language has a complete control structure as well as immediate-mode
+operation.  Functions can be defined and saved for later execution.
+.PP
+Two five hundred-digit numbers can be multiplied to give a
+thousand digit result in about ten seconds.
+.PP
+A small collection of library functions is also available,
+including sin, cos, arctan, log, exponential, and Bessel functions of
+integer order.
+.PP
+Some of the uses of this compiler are
+.IP \-
+to do computation with large integers,
+.IP \-
+to do computation accurate to many decimal places,
+.IP \-
+conversion of numbers from one base to another base.
+.AE
+.PP
+.SH
+Introduction
+.PP
+BC is a language and a compiler for doing arbitrary precision
+arithmetic on the UNIX time-sharing system [1].
+The compiler was written to make conveniently available a
+collection of routines (called DC [6]) which are capable of doing
+arithmetic on integers of arbitrary size.  The compiler
+is by no means intended to provide a complete programming
+language.
+It is a minimal language facility.
+.PP
+There is a scaling provision that permits the
+use of decimal point notation.
+Provision is made for input and output in bases other than
+decimal.  Numbers can be converted from decimal to octal by
+simply setting the output base to equal 8.
+.PP
+The actual limit on the number of digits that can
+be handled depends on the amount of storage available on the machine.
+Manipulation of numbers with many hundreds of digits
+is possible even on the smallest versions of UNIX.
+.PP
+The syntax of BC has been deliberately selected to agree
+substantially with the C language [2,3].  Those who
+are familiar with C will find few surprises in this language.
+.SH
+Simple Computations with Integers
+.PP
+The simplest kind of statement is an arithmetic expression
+on a line by itself.
+For instance, if you type in the line:
+.DS
+142857 + 285714
+.DE
+the program responds immediately with the line
+.DS
+428571
+.DE
+The operators \-, *, /, %, and ^ can also be used; they
+indicate subtraction, multiplication, division, remaindering, and
+exponentiation, respectively.  Division of integers produces an
+integer result truncated toward zero.
+Division by zero produces an error
+comment.
+.PP
+Any term in an expression may be prefixed by a minus sign to
+indicate that it is to be negated (the `unary' minus sign).
+The expression
+.DS
+7+\-3
+.DE
+is interpreted to mean that \-3 is to be added to 7.
+.PP
+More complex expressions with several operators and with
+parentheses are interpreted just as in
+Fortran, with ^ having the greatest binding
+power, then * and % and /, and finally + and \-.
+Contents of parentheses are evaluated before material
+outside the parentheses.
+Exponentiations are
+performed from right to left and the other operators
+from left to right.
+The two expressions
+.DS
+a^b^c  and  a^(b^c)
+.DE
+are equivalent, as are the two expressions
+.DS
+a*b*c  and  (a*b)*c
+.DE
+BC shares with Fortran and C the undesirable convention that
+.DS
+a/b*c  is equivalent to  (a/b)*c
+.DE
+.PP
+Internal storage registers to hold numbers have single lower-case
+letter names.  The value of an expression can be assigned to
+a register in the usual way.  The statement
+.DS
+x = x + 3
+.DE
+has the effect of increasing by three the value of the contents of the
+register named x.
+When, as in this case, the outermost operator is an =, the
+assignment is performed but the result is not printed.
+Only 26 of these named storage registers are available.
+.PP
+There is a built-in square root function whose
+result is truncated to an integer (but see scaling below).
+The lines
+.DS
+x = sqrt(191)
+x
+.DE
+produce the printed result
+.DS
+13
+.DE
+.SH
+Bases
+.PP
+There are special internal quantities, called `ibase' and `obase'.
+The contents of `ibase', initially set to 10,
+determines the base used for interpreting numbers read in.
+For example, the lines
+.DS
+ibase = 8
+11
+.DE
+will produce the output line
+.DS
+9
+.DE
+and you are all set up to do octal to decimal conversions.
+Beware, however of trying to change the input base back
+to decimal by typing
+.DS
+ibase = 10
+.DE
+Because the number 10 is interpreted as octal, this statement will
+have no effect.
+For those who deal in hexadecimal notation,
+the characters A\-F are permitted in numbers
+(no matter what base is in effect)
+and are
+interpreted as digits having values 10\-15 respectively.
+The statement
+.DS
+ibase = A
+.DE
+will change you back to decimal input base no matter what the
+current input base is.
+Negative and large positive input bases are
+permitted but useless.
+No mechanism has been provided for the input of arbitrary
+numbers in bases less than 1 and greater than 16.
+.PP
+The contents of `obase', initially set to 10, are used as the base for output
+numbers.  The lines
+.DS
+obase = 16
+1000
+.DE
+will produce the output line
+.DS
+3E8
+.DE
+which is to be interpreted as a 3-digit hexadecimal number.
+Very large output bases are permitted, and they are sometimes useful.
+For example, large numbers can be output in groups of five digits
+by setting `obase' to 100000.
+Strange (i.e. 1, 0, or negative) output bases are
+handled appropriately.
+.PP
+Very large numbers are split across lines with 70 characters per line.
+Lines which are continued end with \\.
+Decimal output conversion is practically instantaneous, but output
+of very large numbers (i.e., more than 100 digits) with other bases
+is rather slow.
+Non-decimal output conversion of
+a one hundred digit number takes about
+three seconds.
+.PP
+It is best to remember that `ibase' and `obase' have no effect
+whatever on the course of internal computation or
+on the evaluation of expressions, but only affect input and
+output conversion, respectively.
+.SH
+Scaling
+.PP
+A third special internal quantity called `scale' is
+used to determine the scale of calculated
+quantities.
+Numbers may have
+up to 99 decimal digits after the decimal point.
+This fractional part is retained in further computations.
+We refer to the number of digits after the decimal point of
+a number as its scale.
+.PP
+When two scaled numbers are combined by
+means of one of the arithmetic operations, the result
+has a scale determined by the following rules.  For
+addition and subtraction, the scale of the result is the larger
+of the scales of the two operands.  In this case,
+there is never any truncation of the result.
+For multiplications, the scale of the result is never
+less than the maximum of the two scales of the operands,
+never more than the sum of the scales of the operands
+and, subject to those two restrictions,
+the scale of the result is set equal to the contents of the internal
+quantity `scale'.
+The scale of a quotient is the contents of the internal
+quantity `scale'.  The scale of a remainder is
+the sum of the scales of the quotient and the divisor.
+The result of an exponentiation is scaled as if
+the implied multiplications were performed.
+An exponent must be an integer.
+The scale of a square root is set to the maximum of the scale
+of the argument and the contents of `scale'.
+.PP
+All of the internal operations are actually carried out in terms
+of integers, with digits being discarded when necessary.
+In every case where digits are discarded, truncation and
+not rounding is performed.
+.PP
+The contents of
+`scale' must be no greater than
+99 and no less than 0.  It is initially set to 0.
+In case you need more than 99 fraction digits, you may arrange
+your own scaling.
+.PP
+The internal quantities `scale', `ibase', and `obase' can be
+used in expressions just like other variables.
+The line
+.DS
+scale = scale + 1
+.DE
+increases the value of `scale' by one, and the line
+.DS
+scale
+.DE
+causes the current value of `scale' to be printed.
+.PP
+The value of `scale' retains its meaning as a
+number of decimal digits to be retained in internal
+computation even when `ibase' or `obase' are not equal to 10.
+The internal computations (which are still conducted in decimal,
+regardless of the bases) are performed to the specified number
+of decimal digits, never hexadecimal or octal or any
+other kind of digits.
+.SH
+Functions
+.PP
+The name of a function is a single lower-case letter.
+Function names are permitted to collide with simple
+variable names.
+Twenty-six different defined functions are permitted
+in addition to the twenty-six variable names.
+The line
+.DS
+       define a(x){
+.DE
+begins the definition of a function with one argument.
+This line must be followed by one or more statements,
+which make up the body of the function, ending
+with a right brace }.
+Return of control from a function occurs when a return
+statement is executed or when the end of the function is reached.
+The return statement can take either
+of the two forms
+.DS
+return
+return(x)
+.DE
+In the first case, the value of the function is 0, and in
+the second, the value of the expression in parentheses.
+.PP
+Variables used in the function can be declared as automatic
+by a statement of the form
+.DS
+auto x,y,z
+.DE
+There can be only one `auto' statement in a function and it must
+be the first statement in the definition.
+These automatic variables are allocated space and initialized
+to zero on entry to the function and thrown away on return.  The
+values of any variables with the same names outside the function
+are not disturbed.
+Functions may be called recursively and the automatic variables
+at each level of call are protected.
+The parameters named in a function definition are treated in
+the same way as the automatic variables of that function
+with the single exception that they are given a value
+on entry to the function.
+An example of a function definition is
+.DS
+       define a(x,y){
+               auto z
+               z = x*y
+               return(z)
+       }
+.DE
+The value of this function, when called, will be the
+product of its
+two arguments.
+.PP
+A function is called by the appearance of its name
+followed by a string of arguments enclosed in
+parentheses and separated by commas.
+The result
+is unpredictable if the wrong number of arguments is used.
+.PP
+Functions with no arguments are defined and called using
+parentheses with nothing between them: b().
+.PP
+If the function
+.ft I
+a
+.ft
+above has been defined, then the line
+.DS
+a(7,3.14)
+.DE
+would cause the result 21.98 to be printed and the line
+.DS
+x = a(a(3,4),5)
+.DE
+would cause the value of x to become 60.
+.SH
+Subscripted Variables
+.PP
+A single lower-case letter variable name
+followed by an expression in brackets is called a subscripted
+variable (an array element).
+The variable name is called the array name and the expression
+in brackets is called the subscript.
+Only one-dimensional arrays are
+permitted.  The names of arrays are permitted to
+collide with the names of simple variables and function names.
+Any fractional
+part of a subscript is discarded before use.
+Subscripts must be greater than or equal to zero and 
+less than or equal to 2047.
+.PP
+Subscripted variables may be freely used in expressions, in
+function calls, and in return statements.
+.PP
+An array name may be used as an argument to a function,
+or may be declared as automatic in
+a function definition by the use of empty brackets:
+.DS
+f(a[\|])
+define f(a[\|])
+auto a[\|]
+.DE
+When an array name is so used, the whole contents of the array
+are copied for the use of the function, and thrown away on exit
+from the function.
+Array names which refer to whole arrays cannot be used
+in any other contexts.
+.SH
+Control Statements
+.PP
+The `if', the `while', and the `for' statements
+may be used to alter the flow within programs or to cause iteration.
+The range of each of them is a statement or
+a compound statement consisting of a collection of
+statements enclosed in braces.
+They are written in the following way
+.DS
+if(relation) statement
+while(relation) statement
+for(expression1; relation; expression2) statement
+.DE
+or
+.DS
+if(relation) {statements}
+while(relation) {statements}
+for(expression1; relation; expression2) {statements}
+.DE
+.PP
+A relation in one of the control statements is an expression of the form
+.DS
+x>y
+.DE
+where  two expressions are related by one of the six relational
+operators <, >, <=, >=, ==, or !=.
+The relation ==
+stands for `equal to' and != stands for `not equal to'.
+The meaning of the remaining relational operators is
+clear.
+.PP
+BEWARE of using = instead of == in a relational.  Unfortunately,
+both of them are legal, so you will not get a diagnostic
+message, but = really will not do a comparison.
+.PP
+The `if' statement causes execution of its range
+if and only if the relation is true.
+Then control passes to the next statement in sequence.
+.PP
+The `while' statement causes execution of its range
+repeatedly as long as the relation
+is true.  The relation is tested before each execution
+of its range and if the relation
+is false, control passes to the next statement beyond the range
+of the while.
+.PP
+The `for' statement begins
+by executing `expression1'.  Then the relation is tested
+and, if true, the statements in the range of the `for' are executed.
+Then `expression2' is executed.  The relation is tested, and so on.
+The typical use of the `for' statement is for a controlled iteration,
+as in the statement
+.DS
+for(i=1; i<=10; i=i+1) i
+.DE
+which will print the integers from 1 to 10.
+Here are some examples of the use of the control statements.
+.DS
+define f(n){
+auto i, x
+x=1
+for(i=1; i<=n; i=i+1) x=x*i
+return(x)
+}
+.DE
+The line
+.DS
+       f(a)
+.DE
+will print
+.ft I
+a
+.ft
+factorial if
+.ft I
+a
+.ft
+is a positive integer.
+Here is the definition of a function which will
+compute values of the binomial coefficient
+(m and n are assumed to be positive integers).
+.DS
+define b(n,m){
+auto x, j
+x=1
+for(j=1; j<=m; j=j+1) x=x*(n\-j+1)/j
+return(x)
+}
+.DE
+The following function computes values of the exponential function
+by summing the appropriate series
+without regard for possible truncation errors:
+.DS
+scale = 20
+define e(x){
+       auto a, b, c, d, n
+       a = 1
+       b = 1
+       c = 1
+       d = 0
+       n = 1
+       while(1==1){
+               a = a*x
+               b = b*n
+               c = c + a/b
+               n = n + 1
+               if(c==d) return(c)
+               d = c
+       }
+}
+.DE
+.SH
+Some Details
+.PP
+There are some language features that every user should know
+about even if he will not use them.
+.PP
+Normally statements are typed one to a line.  It is also permissible
+to type several statements on a line separated by semicolons.
+.PP
+If an assignment statement is parenthesized, it then has
+a value and it can be used anywhere that an expression can.
+For example, the line
+.DS
+(x=y+17)
+.DE
+not only makes the indicated assignment, but also prints the
+resulting value.
+.PP
+Here is an example of a use of the value of an
+assignment statement even when it is not parenthesized.
+.DS
+x = a[i=i+1]
+.DE
+causes a value to be assigned to x and also increments i
+before it is used as a subscript.
+.PP
+The following constructs work in BC in exactly the same manner
+as they do in the C language.  Consult the appendix or the
+C manuals [2,3] for their exact workings.
+.DS
+.ta 2i
+x=y=z  is the same as  x=(y=z)
+x =+ y x = x+y
+x =\- y        x = x\-y
+x =* y x = x*y
+x =/ y x = x/y
+x =% y x = x%y
+x =^ y x = x^y
+x++    (x=x+1)\-1
+x\-\-  (x=x\-1)+1
+++x    x = x+1
+\-\-x  x = x\-1
+.DE
+Even if you don't intend to use the constructs,
+if you type one inadvertently, something correct but unexpected
+may happen.
+.PP
+WARNING!  In some of these constructions, spaces are
+significant.
+There is a real difference between
+x=\-y and x= \-y.
+The first replaces x by x\-y and the second by \-y.
+.SH
+Three Important Things
+.PP
+1.  To exit a BC program, type `quit'.
+.PP
+2. There is a comment convention identical to that of C and
+of PL/I.  Comments begin with `/*' and end with `*/'.
+.PP
+3. There is a library of math functions which may be obtained by
+typing at command level
+.DS
+bc \-l
+.DE
+This command will load a set of library functions
+which, at the time of writing, consists of sine (named `s'),
+cosine (`c'), arctangent (`a'), natural logarithm (`l'),
+exponential (`e') and Bessel functions of integer order (`j(n,x)').  Doubtless more functions will be added
+in time.
+The library sets the scale to 20.  You can reset it to something
+else if you like.
+The design of these mathematical library routines
+is discussed elsewhere [4].
+.PP
+If you type
+.DS
+bc file ...
+.DE
+BC will read and execute the named file or files before accepting
+commands from the keyboard.  In this way, you may load your
+favorite programs and function definitions.
+.SH
+Acknowledgement
+.PP
+The compiler is written in YACC [5]; its original
+version  was written by S. C. Johnson.
+.SH
+References
+.IP[1]
+K. Thompson and D. M. Ritchie,
+.ft I
+UNIX Programmer's Manual,
+.ft
+Fifth Edition (1974)
+.IP[2]
+D. M. Ritchie,
+.ft I
+C Reference Manual,
+.ft
+.IP[3]
+B. W. Kernighan,
+.ft I
+Programming in C: A Tutorial,
+.ft
+.IP[4]
+Robert Morris,
+.ft I
+A Library of Reference Standard Mathematical Subroutines,
+.ft
+.IP[5]
+S. C. Johnson,
+.ft I
+YACC, Yet Another Compiler-Compiler,
+.ft
+.IP[6]
+R. Morris and L. L. Cherry,
+.ft I
+DC \- An Interactive Desk Calculator,
+.ft
+.LP

diff --git a/usr/doc/bc/bca b/usr/doc/bc/bca
new file mode 100644 (file)
index 0000000..6aab857
--- /dev/null
+++ b/usr/doc/bc/bca
@@ -0,0 +1,440 @@
+.bp
+.ft B
+.DS C
+Appendix
+.DE
+.ft
+.NH
+Notation
+.PP
+In the following pages syntactic categories are in \fIitalics\fP;
+literals are in \fBbold\fP; material in brackets [\|] is optional.
+.NH
+Tokens
+.PP
+Tokens consist of keywords, identifiers, constants, operators,
+and separators.
+Token separators may be blanks, tabs or comments.
+Newline characters or semicolons separate statements.
+.NH 2
+Comments
+.PP
+Comments are introduced by the characters /* and terminated by
+*/.
+.NH 2
+Identifiers
+.PP
+There are three kinds of identifiers \- ordinary identifiers, array identifiers
+and function identifiers.
+All three types consist of single lower-case letters.
+Array identifiers are followed by square brackets, possibly
+enclosing an expression describing a subscript.
+Arrays are singly dimensioned and may contain up to 2048
+elements.
+Indexing begins at zero so an array may be indexed from 0 to 2047.
+Subscripts are truncated to integers.
+Function identifiers are followed by parentheses, possibly enclosing arguments.
+The three types of identifiers do not conflict;
+a program can have a variable named \fBx\fP,
+an array named \fBx\fP and a function named \fBx\fP, all of which are separate and
+distinct.
+.NH 2
+Keywords
+.PP
+The following are reserved keywords:
+.ft B
+.ta .5i 1.0i
+.nf
+       ibase   if
+       obase   break
+       scale   define
+       sqrt    auto
+       length  return
+       while   quit
+       for
+.fi
+.ft
+.NH 2
+Constants
+.PP
+Constants consist of arbitrarily long numbers
+with an optional decimal point.
+The hexadecimal digits \fBA\fP\-\fBF\fP are also recognized as digits with
+values 10\-15, respectively.
+.NH 1
+Expressions
+.PP
+The value of an expression is printed unless the main
+operator is an assignment.
+Precedence is the same as the order
+of presentation here, with highest appearing first.
+Left or right associativity, where applicable, is
+discussed with each operator.
+.bp
+.NH 2
+Primitive expressions
+.NH 3
+Named expressions
+.PP
+Named expressions are
+places where values are stored.
+Simply stated,
+named expressions are legal on the left
+side of an assignment.
+The value of a named expression is the value stored in the place named.
+.NH 4
+\fIidentifiers\fR
+.PP
+Simple identifiers are named expressions.
+They have an initial value of zero.
+.NH 4
+\fIarray-name\fP\|[\|\fIexpression\fP\|]
+.PP
+Array elements are named expressions.
+They have an initial value of zero.
+.NH 4
+\fBscale\fR, \fBibase\fR and \fBobase\fR
+.PP
+The internal registers
+\fBscale\fP, \fBibase\fP and \fBobase\fP are all named expressions.
+\fBscale\fP is the number of digits after the decimal point to be
+retained in arithmetic operations.
+\fBscale\fR has an initial value of zero.
+\fBibase\fP and \fBobase\fP are the input and output number
+radix respectively.
+Both \fBibase\fR and \fBobase\fR have initial values of 10.
+.NH 3
+Function calls
+.NH 4
+\fIfunction-name\fB\|(\fR[\fIexpression\fR\|[\fB,\|\fIexpression\|\fR.\|.\|.\|]\|]\fB)
+.PP
+A function call consists of a function name followed by parentheses
+containing a comma-separated list of
+expressions, which are the function arguments.
+A whole array passed as an argument is specified by the
+array name followed by empty square brackets.
+All function arguments are passed by
+value.
+As a result, changes made to the formal parameters have
+no effect on the actual arguments.
+If the function terminates by executing a return
+statement, the value of the function is
+the value of the expression in the parentheses of the return
+statement or is zero if no expression is provided
+or if there is no return statement.
+.NH 4
+sqrt\|(\|\fIexpression\fP\|)
+.PP
+The result is the square root of the expression.
+The result is truncated in the least significant decimal place.
+The scale of the result is
+the scale of the expression or the
+value of
+.ft B
+scale,
+.ft
+whichever is larger.
+.NH 4
+length\|(\|\fIexpression\fP\|)
+.PP
+The result is the total number of significant decimal digits in the expression.
+The scale of the result is zero.
+.NH 4
+scale\|(\|\fIexpression\fP\|)
+.PP
+The result is the scale of the expression.
+The scale of the result is zero.
+.NH 3
+Constants
+.PP
+Constants are primitive expressions.
+.NH 3
+Parentheses
+.PP
+An expression surrounded by parentheses is
+a primitive expression.
+The parentheses are used to alter the
+normal precedence.
+.NH 2
+Unary operators
+.PP
+The unary operators
+bind right to left.
+.NH 3
+\-\|\fIexpression\fP
+.PP
+The result is the negative of the expression.
+.NH 3
+++\|\fInamed-expression\fP
+.PP
+The named expression is
+incremented by one.
+The result is the value of the named expression after
+incrementing.
+.NH 3
+\-\-\|\fInamed-expression\fP
+.PP
+The named expression is
+decremented by one.
+The result is the value of the named expression after
+decrementing.
+.NH 3
+\fInamed-expression\fP\|++
+.PP
+The named expression is
+incremented by one.
+The result is the value of the named expression before
+incrementing.
+.NH 3
+\fInamed-expression\fP\|\-\-
+.PP
+The named expression is
+decremented by one.
+The result is the value of the named expression before
+decrementing.
+.NH 2
+Exponentiation operator
+.PP
+The exponentiation operator binds right to left.
+.NH 3
+\fIexpression\fP ^ \fIexpression\fP
+.PP
+The result is the first
+expression raised to the power of the
+second expression.
+The second expression must be an integer.
+If \fIa\fP
+is the scale of the left expression
+and \fIb\fP is the absolute value
+of the right expression,
+then the scale of the result is:
+.PP
+min\|(\|\fIa\(mub\fP,\|max\|(\|\fBscale\fP,\|\fIa\fP\|)\|)
+.NH 2
+Multiplicative operators
+.PP
+The operators *, /, % bind left to right.
+.NH 3
+\fIexpression\fP * \fIexpression\fP
+.PP
+The result is the product
+of the two expressions.
+If \fIa\fP and \fIb\fP are the
+scales of the two expressions,
+then the scale of the result is:
+.PP
+min\|(\|\fIa+b\fP,\|max\|(\|\fBscale\fP,\|\fIa\fP,\|\fIb\fP\|)\|)
+.NH 3
+\fIexpression\fP / \fIexpression\fP
+.PP
+The result is the quotient of the two expressions.
+The scale of the result is the value of \fBscale\fR.
+.NH 3
+\fIexpression\fP % \fIexpression\fP
+.PP
+The % operator produces the remainder of the division
+of the two expressions.
+More precisely,
+\fIa\fP%\fIb\fP is \fIa\fP\-\fIa\fP/\fIb\fP*\fIb\fP.
+.PP
+The scale of the result is the sum of the scale of
+the divisor and the value of
+.ft B
+scale
+.ft
+.NH 2
+Additive operators
+.PP
+The additive operators bind left to right.
+.NH 3
+\fIexpression\fP + \fIexpression\fP
+.PP
+The result is the sum of the two expressions.
+The scale of the result is
+the maximun of the scales of the expressions.
+.NH 3
+\fIexpression\fP \- \fIexpression\fP
+.PP
+The result is the difference of the two expressions.
+The scale of the result is the
+maximum of the scales of the expressions.
+.NH 2
+assignment operators
+.PP
+The assignment operators bind right to left.
+.NH 3
+\fInamed-expression\fP = \fIexpression\fP
+.PP
+This expression results in assigning the value of the expression
+on the right
+to the named expression on the left.
+.NH 3
+\fInamed-expression\fP =+ \fIexpression\fP
+.NH 3
+\fInamed-expression\fP =\- \fIexpression\fP
+.NH 3
+\fInamed-expression\fP =* \fIexpression\fP
+.NH 3
+\fInamed-expression\fP =/ \fIexpression\fP
+.NH 3
+\fInamed-expression\fP =% \fIexpression\fP
+.NH 3
+\fInamed-expression\fP =^ \fIexpression\fP
+.PP
+The result of the above expressions is equivalent
+to ``named expression = named expression OP expression'',
+where OP is the operator after the = sign.
+.NH 1
+Relations
+.PP
+Unlike all other operators, the relational operators
+are only valid as the object of an \fBif\fP, \fBwhile\fP,
+or inside a \fBfor\fP statement.
+.NH 2
+\fIexpression\fP < \fIexpression\fP
+.NH 2
+\fIexpression\fP > \fIexpression\fP
+.NH 2
+\fIexpression\fP <= \fIexpression\fP
+.NH 2
+\fIexpression\fP >= \fIexpression\fP
+.NH 2
+\fIexpression\fP == \fIexpression\fP
+.NH 2
+\fIexpression\fP != \fIexpression\fP
+.NH 1
+Storage classes
+.PP
+There are only two storage classes in BC, global and automatic
+(local).
+Only identifiers that are to be local to a function need be 
+declared with the \fBauto\fP command.
+The arguments to a function
+are local to the function.
+All other identifiers are assumed to be global
+and available to all functions.
+All identifiers, global and local, have initial values
+of zero.
+Identifiers declared as \fBauto\fP are allocated on entry to the function 
+and released on returning from the function.
+They therefore do not retain values between function calls.
+\fBauto\fP arrays are specified by the array name followed by empty square brackets.
+.PP
+Automatic variables in BC do not work in exactly the same way
+as in either C or PL/I.  On entry to a function, the old values of
+the names that appear as parameters and as automatic
+variables are pushed onto a stack.  
+Until return is made from the function, reference to these
+names refers only to the new values.
+.NH 1
+Statements
+.PP
+Statements must be separated by semicolon or newline.
+Except where altered by control statements, execution
+is sequential.
+.NH 2
+Expression statements
+.PP
+When a statement is an expression, unless
+the main operator is an assignment, the value
+of the expression is printed, followed by a newline character.
+.NH 2
+Compound statements
+.PP
+Statements may be grouped together and used when one statement is expected
+by surrounding them with { }.
+.NH 2
+Quoted string statements
+.PP
+"any string"
+.sp .5
+This statement prints the string inside the quotes.
+.NH 2
+If statements
+.sp .5
+\fBif\|(\|\fIrelation\fB\|)\|\fIstatement\fR
+.PP
+The substatement is executed if the relation is true.
+.NH 2
+While statements
+.sp .5
+\fBwhile\|(\|\fIrelation\fB\|)\|\fIstatement\fR
+.PP
+The statement is executed while the relation
+is true.
+The test occurs before each execution of the statement.
+.NH 2
+For statements
+.sp .5
+\fBfor\|(\|\fIexpression\fB; \fIrelation\fB; \fIexpression\fB\|)\|\fIstatement\fR
+.PP
+The for statement is the same as
+.nf
+.ft I
+       first-expression
+       \fBwhile\|(\fPrelation\|\fB) {\fP
+               statement
+               last-expression
+       }
+.ft R
+.fi
+.PP
+All three expressions must be present.
+.NH 2
+Break statements
+.sp .5
+\fBbreak\fP
+.PP
+\fBbreak\fP causes termination of a \fBfor\fP or \fBwhile\fP statement.
+.NH 2
+Auto statements
+.sp .5
+\fBauto \fIidentifier\fR\|[\|\fB,\fIidentifier\fR\|]
+.PP
+The auto statement causes the values of the identifiers to be pushed down.
+The identifiers can be ordinary identifiers or array identifiers.
+Array identifiers are specified by following the array name by empty square
+brackets.
+The auto statement must be the first statement
+in a function definition.
+.NH 2
+Define statements
+.sp .5
+.nf
+\fBdefine(\|\fR[\fIparameter\|\fR[\fB\|,\|\fIparameter\|.\|.\|.\|\fR]\|]\|\fB)\|{\fI
+       statements\|\fB}\fR
+.fi
+.PP
+The define statement defines a function.
+The parameters may
+be ordinary identifiers or array names.
+Array names must be followed by empty square brackets.
+.NH 2
+Return statements
+.sp .5
+\fBreturn\fP
+.sp .5
+\fBreturn(\fI\|expression\|\fB)\fR
+.PP
+The return statement causes termination of a function,
+popping of its auto variables, and
+specifies the result of the function.
+The first form is equivalent to \fBreturn(0)\fR.
+The result of the function is the result of the expression
+in parentheses.
+.NH 2
+Quit
+.PP
+The quit statement stops execution of a BC program and returns
+control to UNIX when it is first encountered.
+Because it is not treated as an executable statement,
+it cannot be used
+in a function definition or in an 
+.ft B
+if, for,
+.ft
+or
+.ft B
+while
+.ft
+statement.

diff --git a/usr/doc/bc/dc b/usr/doc/bc/dc
new file mode 100644 (file)
index 0000000..de2674d
--- /dev/null
+++ b/usr/doc/bc/dc
@@ -0,0 +1,695 @@
+.RP 75-1271-8 39199 39199-11
+.TL
+DC \- An Interactive Desk Calculator
+.AU "MH 2C-524" 3878
+Robert Morris
+.AU
+Lorinda Cherry
+.AI
+.MH
+.AB
+DC is an interactive desk calculator program implemented
+on the UNIX time-sharing system to do arbitrary-precision
+integer arithmetic.
+It has provision for manipulating scaled fixed-point numbers and
+for input and output in bases other than decimal.
+.PP
+The size of numbers that can be manipulated is limited
+only by available core storage.
+On typical implementations of UNIX, the size of numbers that
+can be handled varies from several hundred digits on the smallest
+systems to several thousand on the largest.
+.AE
+.PP
+.SH
+.ND
+.PP
+DC is an arbitrary precision arithmetic package implemented
+on the UNIX time-sharing system
+in the form of an interactive desk calculator.
+It works like a stacking calculator using reverse Polish notation.
+Ordinarily DC operates on decimal integers, but one may
+specify an input base, output base, and a number of fractional
+digits to be maintained.
+.PP
+A language called BC [1] has been developed which accepts
+programs written in the familiar style of higher-level
+programming languages and compiles output which is
+interpreted by DC.
+Some of the commands described below were designed
+for the compiler interface and are not easy for a human user
+to manipulate.
+.PP
+Numbers that are typed into DC are put on a push-down
+stack.
+DC commands work by taking the top number or two
+off the stack, performing the desired operation, and pushing the result
+on the stack.
+If an argument is given,
+input is taken from that file until its end,
+then from the standard input.
+.SH
+SYNOPTIC DESCRIPTION
+.PP
+Here we describe the DC commands that are intended
+for use by people.  The additional commands that are
+intended to be invoked by compiled output are
+described in the detailed description.
+.PP
+Any number of commands are permitted on a line.
+Blanks and new-line characters are ignored except within numbers
+and in places where a register name is expected.
+.PP
+The following constructions are recognized:
+.SH
+number
+.IP
+The value of the number is pushed onto the main stack.
+A number is an unbroken string of the digits 0-9
+and the capital letters A\-F which are treated as digits
+with values 10\-15 respectively.
+The number may be preceded by an underscore \*_ to input a
+negative number.
+Numbers may contain decimal points.
+.SH
++  \-  *  %  ^
+.IP
+The
+top two values on the stack are added
+(\fB+\fP),
+subtracted
+(\fB\-\fP),
+multiplied (\fB*\fP),
+divided (\fB/\fP),
+remaindered (\fB%\fP),
+or exponentiated (^).
+The two entries are popped off the stack;
+the result is pushed on the stack in their place.
+The result of a division is an integer truncated toward zero.
+See the detailed description below for the treatment of
+numbers with decimal points.
+An exponent must not have any digits after the decimal point.
+.SH
+s\fIx\fP
+.IP
+The
+top of the main stack is popped and stored into
+a register named \fIx\fP, where \fIx\fP may be any character.
+If
+the
+.ft B
+s
+.ft
+is capitalized,
+.ft I
+x
+.ft
+is treated as a stack and the value is pushed onto it.
+Any character, even blank or new-line, is a valid register name.
+.SH
+l\fIx\fP
+.IP
+The
+value in register
+.ft I
+x
+.ft
+is pushed onto the stack.
+The register
+.ft I
+x
+.ft
+is not altered.
+If the
+.ft B
+l
+.ft
+is capitalized,
+register
+.ft I
+x
+.ft
+is treated as a stack and its top value is popped onto the main stack.
+.LP
+All registers start with empty value which is treated as a zero
+by the command \fBl\fP and is treated as an error by the command \fBL\fP.
+.SH
+.SH
+d
+.IP
+The
+top value on the stack is duplicated.
+.SH
+p
+.IP
+The top value on the stack is printed.
+The top value remains unchanged.
+.SH
+f
+.IP
+All values on the stack and in registers are printed.
+.SH
+x
+.IP
+treats the top element of the stack as a character string,
+removes it from the stack, and
+executes it as a string of DC commands.
+.SH
+[ ... ]
+.IP
+puts the bracketed character string onto the top of the stack.
+.SH
+q
+.IP
+exits the program.
+If executing a string, the recursion level is
+popped by two.
+If
+.ft B
+q
+.ft
+is capitalized,
+the top value on the stack is popped and the string execution level is popped
+by that value.
+.SH
+<\fIx\fP  >\fIx\fP  =\fIx\fP  !<\fIx\fP  !>\fIx\fP  !=\fIx\fP
+.IP
+The
+top two elements of the stack are popped and compared.
+Register
+.ft I
+x
+.ft
+is executed if they obey the stated
+relation.
+Exclamation point is negation.
+.SH
+v
+.IP
+replaces the top element on the stack by its square root.
+The square root of an integer is truncated to an integer.
+For the treatment of numbers with decimal points, see
+the detailed description below.
+.SH
+!
+.IP
+interprets the rest of the line as a UNIX command.
+Control returns to DC when the UNIX command terminates.
+.SH
+c
+.IP
+All values on the stack are popped; the stack becomes empty.
+.SH
+i
+.IP
+The top value on the stack is popped and used as the
+number radix for further input.
+If \fBi\fP is capitalized, the value of
+the input base is pushed onto the stack.
+No mechanism has been provided for the input of arbitrary
+numbers in bases less than 1 or greater than 16.
+.SH
+o
+.IP
+The top value on the stack is popped and used as the
+number radix for further output.
+If \fBo\fP is capitalized, the value of the output
+base is pushed onto the stack.
+.SH
+k
+.IP
+The top of the stack is popped, and that value is used as
+a scale factor
+that influences the number of decimal places
+that are maintained during multiplication, division, and exponentiation.
+The scale factor must be greater than or equal to zero and
+less than 100.
+If \fBk\fP is capitalized, the value of the scale factor
+is pushed onto the stack.
+.SH
+z
+.IP
+The value of the stack level is pushed onto the stack.
+.SH
+?
+.IP
+A line of input is taken from the input source (usually the console)
+and executed.
+.SH
+DETAILED DESCRIPTION
+.SH
+Internal Representation of Numbers
+.PP
+Numbers are stored internally using a dynamic storage allocator.
+Numbers are kept in the form of a string
+of digits to the base 100 stored one digit per byte
+(centennial digits).
+The string is stored with the low-order digit at the
+beginning of the string.
+For example, the representation of 157
+is 57,1.
+After any arithmetic operation on a number, care is taken
+that all digits are in the range 0\-99 and that
+the number has no leading zeros.
+The number zero is represented by the empty string.
+.PP
+Negative numbers are represented in the 100's complement
+notation, which is analogous to two's complement notation for binary
+numbers.
+The high order digit of a negative number is always \-1
+and all other digits are in the range 0\-99.
+The digit preceding the high order \-1 digit is never a 99.
+The representation of \-157 is 43,98,\-1.
+We shall call this the canonical form of a number.
+The advantage of this kind of representation of negative
+numbers is ease of addition.  When addition is performed digit
+by digit, the result is formally correct.  The result need only
+be modified, if necessary, to put it into canonical form.
+.PP
+Because the largest valid digit is 99 and the byte can
+hold numbers twice that large, addition can be carried out
+and the handling of carries done later when
+that is convenient, as it sometimes is.
+.PP
+An additional byte is stored with each number beyond
+the high order digit to indicate the number of
+assumed decimal digits after the decimal point.  The representation
+of .001 is 1,\fI3\fP
+where the scale has been italicized to emphasize the fact that it
+is not the high order digit.
+The value of this extra byte is called the
+.ft B
+scale factor
+.ft
+of the number.
+.SH
+The Allocator
+.PP
+DC uses a dynamic string storage allocator
+for all of its internal storage.
+All reading and writing of numbers internally is done through
+the allocator.
+Associated with each string in the allocator is a four-word header containing pointers
+to the beginning of the string, the end of the string,
+the next place to write, and the next place to read.
+Communication between the allocator and DC
+is done via pointers to these headers.
+.PP
+The allocator initially has one large string on a list
+of free strings.  All headers except the one pointing
+to this string are on a list of free headers.
+Requests for strings are made by size.
+The size of the string actually supplied is the next higher
+power of 2.
+When a request for a string is made, the allocator
+first checks the free list to see if there is
+a string of the desired size.
+If none is found, the allocator finds the next larger free string and splits it repeatedly until
+it has a string of the right size.
+Left-over strings are put on the free list.
+If there are no larger strings,
+the allocator tries to coalesce smaller free strings into
+larger ones.
+Since all strings are the result
+of splitting large strings,
+each string has a neighbor that is next to it in core
+and, if free, can be combined with it to make a string twice as long.
+This is an implementation of the `buddy system' of allocation
+described in [2].
+.PP
+Failing to find a string of the proper length after coalescing,
+the allocator asks the system for more space.
+The amount of space on the system is the only limitation
+on the size and number of strings in DC.
+If at any time in the process of trying to allocate a string, the allocator runs out of
+headers, it also asks the system for more space.
+.PP
+There are routines in the allocator for reading, writing, copying, rewinding,
+forward-spacing, and backspacing strings.
+All string manipulation is done using these routines.
+.PP
+The reading and writing routines
+increment the read pointer or write pointer so that
+the characters of a string are read or written in
+succession by a series of read or write calls.
+The write pointer is interpreted as the end of the
+information-containing portion of a string and a call
+to read beyond that point returns an end-of-string indication.
+An attempt to write beyond the end of a string
+causes the allocator to
+allocate a larger space and then copy
+the old string into the larger block.
+.SH
+Internal Arithmetic
+.PP
+All arithmetic operations are done on integers.
+The operands (or operand) needed for the operation are popped
+from the main stack and their scale factors stripped off.
+Zeros are added or digits removed as necessary to get
+a properly scaled result from the internal arithmetic routine.
+For example, if the scale of the operands is different and decimal
+alignment is required, as it is for
+addition, zeros are appended to the operand with the smaller
+scale.
+After performing the required arithmetic operation,
+the proper scale factor is appended to the end of the number before
+it is pushed on the stack.
+.PP
+A register called \fBscale\fP plays a part
+in the results of most arithmetic operations.
+\fBscale\fP is the bound on the number of decimal places retained in
+arithmetic computations.
+\fBscale\fP may be set to the number on the top of the stack
+truncated to an integer with the \fBk\fP command.
+\fBK\fP may be used to push the value of \fBscale\fP on the stack.
+\fBscale\fP must be greater than or equal to 0 and less than 100.
+The descriptions of the individual arithmetic operations will
+include the exact effect of \fBscale\fP on the computations.
+.SH
+Addition and Subtraction
+.PP
+The scales of the two numbers are compared and trailing
+zeros are supplied to the number with the lower scale to give both
+numbers the same scale.  The number with the smaller scale is
+multiplied by 10 if the difference of the scales is odd.
+The scale of the result is then set to the larger of the scales
+of the two operands.
+.PP
+Subtraction is performed by negating the number
+to be subtracted and proceeding as in addition.
+.PP
+Finally, the addition is performed digit by digit from the
+low order end of the number.  The carries are propagated
+in the usual way.
+The resulting number is brought into canonical form, which may
+require stripping of leading zeros, or for negative numbers
+replacing the high-order configuration 99,\-1 by the digit \-1.
+In any case, digits which are not in the range 0\-99 must
+be brought into that range, propagating any carries or borrows
+that result.
+.SH
+Multiplication
+.PP
+The scales are removed from the two operands and saved.
+The operands are both made positive.
+Then multiplication is performed in
+a digit by digit manner that exactly mimics the hand method
+of multiplying.
+The first number is multiplied by each digit of the second
+number, beginning with its low order digit.  The intermediate
+products are accumulated into a partial sum which becomes the
+final product.
+The product is put into the canonical form and its sign is
+computed from the signs of the original operands.
+.PP
+The scale of the result is set equal to the sum
+of the scales of the two operands.
+If that scale is larger than the internal register
+.ft B
+scale
+.ft
+and also larger than both of the scales of the two operands,
+then the scale of the result is set equal to the largest
+of these three last quantities.
+.SH
+Division
+.PP
+The scales are removed from the two operands.
+Zeros are appended or digits removed from the dividend to make
+the scale of the result of the integer division equal to
+the internal quantity
+\fBscale\fP.
+The signs are removed and saved.
+.PP
+Division is performed much as it would be done by hand.
+The difference of the lengths of the two numbers
+is computed.
+If the divisor is longer than the dividend,
+zero is returned.
+Otherwise the top digit of the divisor is divided into the top
+two digits of the dividend.
+The result is used as the first (high-order) digit of the
+quotient.
+It may turn out be one unit too low, but if it is, the next
+trial quotient will be larger than 99 and this will be
+adjusted at the end of the process.
+The trial digit is multiplied by the divisor and the result subtracted
+from the dividend and the process is repeated to get
+additional quotient digits until the remaining
+dividend is smaller than the divisor.
+At the end, the digits of the quotient are put into
+the canonical form, with propagation of carry as needed.
+The sign is set from the sign of the operands.
+.SH
+Remainder
+.PP
+The division routine is called and division is performed
+exactly as described.  The quantity returned is the remains of the
+dividend at the end of the divide process.
+Since division truncates toward zero, remainders have the same
+sign as the dividend.
+The scale of the remainder is set to 
+the maximum of the scale of the dividend and
+the scale of the quotient plus the scale of the divisor.
+.SH
+Square Root
+.PP
+The scale is stripped from the operand.
+Zeros are added if necessary to make the
+integer result have a scale that is the larger of
+the internal quantity
+\fBscale\fP
+and the scale of the operand.
+.PP
+The method used to compute sqrt(y) is Newton's method
+with successive approximations by the rule
+.EQ
+x sub {n+1} = \(12 ( x sub n + y over x sub n )
+.EN
+The initial guess is found by taking the integer square root
+of the top two digits.
+.SH
+Exponentiation
+.PP
+Only exponents with zero scale factor are handled.  If the exponent is
+zero, then the result is 1.  If the exponent is negative, then
+it is made positive and the base is divided into one.  The scale
+of the base is removed.
+.PP
+The integer exponent is viewed as a binary number.
+The base is repeatedly squared and the result is
+obtained as a product of those powers of the base that
+correspond to the positions of the one-bits in the binary
+representation of the exponent.
+Enough digits of the result
+removed to make the scale of the result the same as if the
+indicated multiplication had been performed.
+.SH
+Input Conversion and Base
+.PP
+Numbers are converted to the internal representation as they are read
+in.
+The scale stored with a number is simply the number of fractional digits input.
+Negative numbers are indicated by preceding the number with a \fB\_\fP.
+The hexadecimal digits A\-F correspond to the numbers 10\-15 regardless of input base.
+The \fBi\fP command can be used to change the base of the input numbers.
+This command pops the stack, truncates the resulting number to an integer,
+and uses it as the input base for all further input.
+The input base is initialized to 10 but may, for example be changed to
+8 or 16 to do octal or hexadecimal to decimal conversions.
+The command \fBI\fP will push the value of the input base on the stack.
+.SH
+Output Commands
+.PP
+The command \fBp\fP causes the top of the stack to be printed.
+It does not remove the top of the stack.
+All of the stack and internal registers can be output
+by typing the command \fBf\fP.
+The \fBo\fP command can be used to change the output base.
+This command uses the top of the stack, truncated to an integer as
+the base for all further output.
+The output base in initialized to 10.
+It will work correctly for any base.
+The command \fBO\fP pushes the value of the output base on the stack.
+.SH
+Output Format and Base
+.PP
+The input and output bases only affect
+the interpretation of numbers on input and output; they have no
+effect on arithmetic computations.
+Large numbers are output with 70 characters per line;
+a \\ indicates a continued line.
+All choices of input and output bases work correctly, although not all are
+useful.
+A particularly useful output base is 100000, which has the effect of
+grouping digits in fives.
+Bases of 8 and 16 can be used for decimal-octal or decimal-hexadecimal
+conversions.
+.SH
+Internal Registers
+.PP
+Numbers or strings may be stored in internal registers or loaded on the stack
+from registers with the commands \fBs\fP and \fBl\fP.
+The command \fBs\fIx\fR pops the top of the stack and
+stores the result in register \fBx\fP.
+\fIx\fP can be any character.
+\fBl\fIx\fR puts the contents of register \fBx\fP on the top of the stack.
+The \fBl\fP command has no effect on the contents of register \fIx\fP.
+The \fBs\fP command, however, is destructive.
+.SH
+Stack Commands
+.PP
+The command \fBc\fP clears the stack.
+The command \fBd\fP pushes a duplicate of the number on the top of the stack
+on the stack.
+The command \fBz\fP pushes the stack size on the stack.
+The command \fBX\fP replaces the number on the top of the stack
+with its scale factor.
+The command \fBZ\fP replaces the top of the stack
+with its length.
+.SH
+Subroutine Definitions and Calls
+.PP
+Enclosing a string in \fB[]\fP pushes the ascii string on the stack.
+The \fBq\fP command quits or in executing a string, pops the recursion levels by two.
+.SH
+Internal Registers \- Programming DC
+.PP
+The load and store
+commands together with \fB[]\fP to store strings, \fBx\fP to execute
+and the testing commands `<', `>', `=', `!<', `!>', `!=' can be used to program
+DC.
+The \fBx\fP command assumes the top of the stack is an string of DC commands
+and executes it.
+The testing commands compare the top two elements on the stack and if the relation holds, execute the register
+that follows the relation.
+For example, to print the numbers 0-9,
+.DS
+[lip1+  si  li10>a]sa
+0si  lax
+.DE
+.SH
+Push-Down Registers and Arrays
+.PP
+These commands were designed for used by a compiler, not by
+people.
+They involve push-down registers and arrays.
+In addition to the stack that commands work on, DC can be thought
+of as having individual stacks for each register.
+These registers are operated on by the commands \fBS\fP and \fBL\fP.
+\fBS\fIx\fR pushes the top value of the main stack onto the stack for
+the register \fIx\fP.
+\fBL\fIx\fR pops the stack for register \fIx\fP and puts the result on the main
+stack.
+The commands \fBs\fP and \fBl\fP also work on registers but not as push-down
+stacks.
+\fBl\fP doesn't effect the top of the
+register stack, and \fBs\fP destroys what was there before.
+.PP
+The commands to work on arrays are \fB:\fP and \fB;\fP.
+\fB:\fIx\fR pops the stack and uses this value as an index into
+the array \fIx\fP.
+The next element on the stack is stored at this index in \fIx\fP.
+An index must be greater than or equal to 0 and
+less than 2048.
+\fB;\fIx\fR is the command to load the main stack from the array \fIx\fP.
+The value on the top of the stack is the index
+into the array \fIx\fP of the value to be loaded.
+.SH
+Miscellaneous Commands
+.PP
+The command \fB!\fP interprets the rest of the line as a UNIX command and passes
+it to UNIX to execute.
+One other compiler command is \fBQ\fP.
+This command uses the top of the stack as the number of levels of recursion to skip.
+.SH
+DESIGN CHOICES
+.PP
+The real reason for the use of a dynamic storage allocator was
+that a general purpose program could be (and in fact has been)
+used for a variety of other tasks.
+The allocator has some value for input and for compiling (i.e.
+the bracket [...] commands) where it cannot be known in advance
+how long a string will be.
+The result was that at a modest
+cost in execution time, all considerations of string allocation
+and sizes of strings were removed from the remainder of the program
+and debugging was made easier.  The allocation method
+used wastes approximately 25% of available space.
+.PP
+The choice of 100 as a base for internal arithmetic
+seemingly has no compelling advantage.  Yet the base cannot
+exceed 127 because of hardware limitations and at the cost
+of 5% in space, debugging was made a great deal easier and
+decimal output was made much faster.
+.PP
+The reason for a stack-type arithmetic design was
+to permit all DC commands from addition to subroutine execution
+to be implemented in essentially the same way.  The result
+was a considerable degree of logical separation of the final
+program into modules with very little communication between
+modules.
+.PP
+The rationale for the lack of interaction between the scale and the bases
+was to provide an understandable means of proceeding after
+a change of base or scale when numbers had already been entered.
+An earlier implementation which had global notions of
+scale and base did not work out well.
+If the value of
+.ft B
+scale
+.ft
+were to be interpreted in the current
+input or output base,
+then a change of base or scale in the midst of a
+computation would cause great confusion in the interpretation
+of the results.
+The current scheme has the advantage that the value of
+the input and output bases
+are only used for input and output, respectively, and they
+are ignored in all other operations.
+The value of
+scale
+is not used for any essential purpose by any part of the program
+and it is used only to prevent the number of
+decimal places resulting from the arithmetic operations from
+growing beyond all bounds.
+.PP
+The design rationale for the choices for the scales of
+the results of arithmetic were that in no case should
+any significant digits be thrown away if, on appearances, the
+user actually wanted them.  Thus, if the user wants
+to add the numbers 1.5 and 3.517, it seemed reasonable to give
+him the result 5.017 without requiring him to unnecessarily
+specify his rather obvious requirements for precision.
+.PP
+On the the other hand, multiplication and exponentiation produce
+results with many more digits than their operands and it
+seemed reasonable to give as a minimum the number of decimal
+places in the operands but not to give more than that
+number of digits
+unless the user asked for them by specifying a value for \fBscale\fP.
+Square root can be handled in just the same way as multiplication.
+The operation of division gives arbitrarily many decimal places
+and there is simply no way to guess how many places the user
+wants.
+In this case only, the user must
+specify a \fBscale\fP to get any decimal places at all.
+.PP
+The scale of remainder was chosen to make it possible
+to recreate the dividend from the quotient and remainder.
+This is easy to implement; no digits are thrown away.
+.SH
+References
+.IP[1]
+L. L. Cherry, R. Morris,
+.ft I
+BC \- An Arbitrary Precision Desk-Calculator Language,
+.ft
+.IP[2]
+K. C. Knowlton,
+.ft I
+A Fast Storage Allocator,
+.ft
+Comm. ACM \fB8\fP, pp. 623-625 (Oct. 1965)
+.LP