








               DC -- An Interactive Desk Calculator


                          Robert Morris
                         Lorinda Cherry


                            ABSTRACT

          DC  is  an  interactive  desk  calculator  program
     implemented on the UNIX(R) time-sharing  system  to  do
     arbitrary-precision  integer arithmetic.  It has provi-
     sion for manipulating scaled  fixed-point  numbers  and
     for input and output in bases other than decimal.

          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.





     Editor's note: the description of the implementation details
of DC in this paper is only valid for the original version of DC.
The current version of DC uses _a different approach.

     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 inte-
gers,  but one may specify an input base, output base, and a num-
ber of fractional digits to be maintained.

     A language called BC [1] has been  developed  which  accepts
programs  written  in the familiar style of higher-level program-
ming languages and compiles output which is  interpreted  by  DC.
Some  of  the commands described below were designed for the com-
piler interface and are not easy for a human user to  manipulate.

     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.













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SYNOPTIC DESCRIPTION

     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 descrip-
tion.

     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.

     The following constructions are recognized:

number

     The value of the number is pushed onto the  main  stack.   A
     number is an unbroken string of the digits 0-9 and the capi-
     tal letters A-F which are  treated  as  digits  with  values
     10-15 respectively.  The number may be preceded by an under-
     score _ to input a negative  number.   Numbers  may  contain
     decimal points.

++  --  **  %%  ^^

     The  top  two  values on the stack are added (++), subtracted
     (--), multiplied (**), divided (//), remaindered (%%), or  expo-
     nentiated  (^).   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 num-
     bers  with  decimal  points.   An exponent must not have any
     digits after the decimal point.

ss_x

     The top of the main stack is popped and stored into a regis-
     ter named _x, where _x may be any character.  If the ss is cap-
     italized, _x is treated as a stack and the  value  is  pushed
     onto  it.  Any character, even blank or new-line, is a valid
     register name.

ll_x

     The value in register _x is pushed onto the stack.  The  reg-
     ister _x is not altered.  If the ll is capitalized, register _x
     is treated as a stack and its top value is popped  onto  the
     main stack.

All  registers  start with empty value which is treated as a zero
by the command ll and is treated as an error by the command LL.













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dd

     The top value on the stack is duplicated.

pp

     The top value on  the  stack  is  printed.   The  top  value
     remains unchanged.

ff

     All values on the stack and in registers are printed.

xx

     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.

[[ ...... ]]

     puts  the  bracketed  character  string  onto the top of the
     stack.

qq

     exits the program.  If executing  a  string,  the  recursion
     level  is popped by two.  If qq is capitalized, the top value
     on the stack is popped and the  string  execution  level  is
     popped by that value.

<<_x  >>_x  ==_x  !!<<_x  !!>>_x  !!==_x

     The  top  two elements of the stack are popped and compared.
     Register _x is executed if they  obey  the  stated  relation.
     Exclamation point is negation.

vv

     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.

!!

     interprets the rest of the line as a UNIX command.   Control
     returns to DC when the UNIX command terminates.

cc

     All values on the stack are popped; the stack becomes empty.











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ii

     The top value on the stack is popped and used as the  number
     radix  for further input.  If ii 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.

oo

     The top value on the stack is popped and used as the  number
     radix for further output.  If oo is capitalized, the value of
     the output base is pushed onto the stack.

kk

     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 kk is  capitalized,  the
     value of the scale factor is pushed onto the stack.

zz

     The value of the stack level is pushed onto the stack.

??

     A  line of input is taken from the input source (usually the
     console) and executed.

DETAILED DESCRIPTION

Internal Representation of Numbers

     Numbers are stored internally using a dynamic storage  allo-
cator.  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 dig-
its are in the range 0-99 and that  the  number  has  no  leading
zeros.  The number zero is represented by the empty string.

     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 representa-
tion 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









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only be modified, if necessary, to put it into canonical form.

     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.

     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,_3 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 scale factor of the number.

The Allocator

     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.

     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].

     Failing to find a string of the proper length after coalesc-
ing, 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.

     There are routines in the allocator  for  reading,  writing,
copying,  rewinding,  forward-spacing,  and  backspacing strings.
All string manipulation is done using these routines.

     The reading and writing routines increment the read  pointer
or  write  pointer so that the characters of a string are read or









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written in succession by a series of read or  write  calls.   The
write  pointer  is interpreted as the end of the information-con-
taining 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.

Internal Arithmetic

     All  arithmetic  operations are done on integers.  The oper-
ands (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  opera-
tion,  the proper scale factor is appended to the end of the num-
ber before it is pushed on the stack.

     A register called scale plays a part in the results of  most
arithmetic operations.  scale is the bound on the number of deci-
mal places retained in arithmetic computations.  scale may be set
to  the  number  on  the top of the stack truncated to an integer
with the kk command.  KK may be used to push the value of scale  on
the  stack.   scale  must  be greater than or equal to 0 and less
than 100.  The descriptions of the individual  arithmetic  opera-
tions will include the exact effect of scale on the computations.

Addition and Subtraction

     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.

     Subtraction  is  performed by negating the number to be sub-
tracted and proceeding as in addition.

     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.

Multiplication

     The scales are removed from the two operands and saved.  The
operands  are  both  made  positive.   Then   multiplication   is









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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.

     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  scale 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.

Division

     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
scale.  The signs are removed and saved.

     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 div-
idend and the process is repeated to get additional quotient dig-
its 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.

Remainder

     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 divi-
dend.   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.

Square Root

     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 scale and the scale of the oper-
and.











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     The method used to compute sqrt(y) is Newton's  method  with
successive approximations by the rule

        x sub {n+1} ~=~ half ( x sub n + y over x sub n )

The  initial  guess is found by taking the integer square root of
the top two digits.

Exponentiation

     Only exponents with zero scale factor are handled.   If  the
exponent is zero, then the result is 1.  If the exponent is nega-
tive, then it is made positive and the base is divided into  one.
The scale of the base is removed.

     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 are removed to make the scale of the result
the same as if the indicated multiplication had been performed.

Input Conversion and Base

     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 __ (an underscore).  The hexadecimal
digits A-F correspond to the numbers 10-15  regardless  of  input
base.   The ii 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 fur-
ther input.  The input base is initialized to  10  but  may,  for
example be changed to 8 or 16 to do octal or hexadecimal to deci-
mal conversions.  The command II will push the value of the  input
base on the stack.

Output Commands

     The command pp 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 ff.  The oo
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 OO pushes the value of
the output base on the stack.

Output Format and Base

     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









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output bases work correctly, although not all are useful.  A par-
ticularly  useful  output base is 100000, which has the effect of
grouping digits in fives.  Bases of 8 and 16 can be used for dec-
imal-octal or decimal-hexadecimal conversions.

Internal Registers

     Numbers  or  strings  may be stored in internal registers or
loaded on the stack from registers with the  commands  ss  and  ll.
The command ss_x pops the top of the stack and stores the result in
register xx.  _x can be any character.  ll_x  puts  the  contents  of
register  xx on the top of the stack.  The ll command has no effect
on the contents of  register  _x.   The  ss  command,  however,  is
destructive.

Stack Commands

     The  command  cc  clears  the  stack.  The command dd pushes a
duplicate of the number on the top of the  stack  on  the  stack.
The  command zz pushes the stack size on the stack.  The command XX
replaces the number on the top of the stack with its  scale  fac-
tor.   The  command  ZZ  replaces  the  top  of the stack with its
length.

Subroutine Definitions and Calls

     Enclosing a string in [[ ]] pushes the  ascii  string  on  the
stack.   The  qq  command quits or in executing a string, pops the
recursion levels by two.

Internal Registers -- Programming DC

     The load and store commands  together  with  [[  ]]  to  store
strings,  xx  to  execute  and the testing commands `<', `>', `=',
`!<', `!>', `!=' can be  used  to  program  DC.   The  xx  command
assumes the top of the stack is an string of DC commands and exe-
cutes 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,

     [lip1+  si  li10>a]sa
     0si  lax


Push-Down Registers and Arrays

     These commands were designed for used by a compiler, not  by
people.   They  involve push-down registers and arrays.  In addi-
tion 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 SS and LL.  SS_x pushes the top value  of
the  main  stack  onto the stack for the register _x.  LL_x pops the
stack for register _x and puts the result on the main stack.   The
commands  ss  and  ll  also  work on registers but not as push-down









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stacks.  ll doesn't effect the top of the register  stack,  and  ss
destroys what was there before.

     The  commands  to  work  on arrays are :: and ;;.  ::_x pops the
stack and uses this value as an index into the array _x.  The next
element on the stack is stored at this index in _x.  An index must
be greater than or equal to 0 and less than 2048.  ;;_x is the com-
mand  to  load the main stack from the array _x.  The value on the
top of the stack is the index into the array _x of the value to be
loaded.

Miscellaneous Commands

     The command !! interprets the rest of the line as a UNIX com-
mand and passes it to UNIX to execute.  One other  compiler  com-
mand  is QQ.  This command uses the top of the stack as the number
of levels of recursion to skip.

DESIGN CHOICES

     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 [...] com-
mands) 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.

     The  choice  of  100 as a base for internal arithmetic seem-
ingly 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.

     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 con-
siderable degree of logical separation of the final program  into
modules with very little communication between modules.

     The  rationale for the lack of interaction between the scale
and the bases was to provide an understandable means of  proceed-
ing 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 scale 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  cur-
rent  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









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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.

     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.

     On 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 scale.
Square root can be handled in just the same  way  as  multiplica-
tion.   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 scale to
get any decimal places at all.

     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.

References

[1]  L. L. Cherry, R. Morris, BC _- An Arbitrary  Precision  Desk-
     Calculator Language.

[2]  K.  C.  Knowlton, _A Fast Storage Allocator, Comm. ACM 88, pp.
     623-625 (Oct. 1965).

























