cupl(1) Manual Page

                          Appendix A

                        Summary of CUPL


ELEMENTS OF THE LANGUAGE

     Characters
     ----------
          Letters: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
          Digits:  0 1 2 3 4 5 6 7 8 9
          Special Characters:  + - * / ( ) . , ' =

     Numbers
     -------
          Normal decimal usage:  e.g. 3, 1.725, -.06 .
          "Scientific" notation:  -1.2E-3  for -1.2 x 10^3 .
          Truncated to 9 significant figures by the system.
          Range:  Absolute values from 10^-78 to 10^76, and 0 .

     Variables and Labels
     --------------------
          a.  Consist of 1 to 8 letters or digits, beginning with
              a letter; no blanks or special characters.

          b.  Must not be one of the following "reserved words":

                 ABS        DOT     IF    OR        THEN
                 ALL        ELSE    INV   PERFORM   TIMES
                 ALLOCATE   END     LE    POSMAX    TO
                 AND        EXP     LET   POSMIN    TRC
                 ATAN       FLOOR   LN    RAND      TRN
                 BLOCK      FOR     LOG   READ      WATCH
                 BY         GE      LT    SGM       WHILE
                 COMMENT    GO      MAX   SIN       WRITE
                 COS        GT      MIN   SQRT
                 DET        IDN     NE    STOP

          c.  Must be unique--same name cannot be used for both a
              label and a variable.

          d.  Variables have automatic initial value of zero.

          e.  Variables can be scalar, vector, or matrix.  [See 8-3 and 9-8]

                If A   is a matrix variable:

                   A        refers to the entire matrix.
                   A(I,J)   refers to the element at the
                              intersection of the ith row
                              and the jth column
                   A(*,J)   refers to the jth column [see 9-2]
                   A(I,*)   refers to the ith row    [see 9-2]
^L
                          A-2

                If V is a vector variable:

                   V refers to the entire vector
                   V(I) refers to the ith component.

              [See 11-3 for subscript round-off rules]

          f.  A vector is a 1-column matrix

          g. IDN is the identity matrix of any appropriate size.

     Arithmetic Operators
     --------------------
          a.  +, -, /, * for multiplication, ** for exponentiation

          b.  Normal precedence:  **, * and /, + and -
                                  Parentheses from inner to outer.
                                  Sequence of + and - from left to
                                  right.

          c. Types of operand:    +, -, /, *, ** for scalars
                                  +, -, *, and numerator of / for
                                     vectors and matrices.

     Spacing
     -------
          No spaces, or splitting at end of line, in any number,
          variable, label, reserved word, or **.  Spaes allowable
          anywhere else.

     Functions
     ---------
          Form                Action              Type of Argument
          ----                ------              ----------------
          ABS(a)        absolute value of a      any expression
          ATAN(a)       arctangent of a, in      numerical valued
                          radians                  expression
          COS(a)        cosine of a, a in        numerical valued
                          radians                  expression
          EXP(e)        e raised to a power      numerical valued
                                                   expression
          FLOOR(a)      greatest integer not     numerical valued
                          exceeding a              expression
          LN(a)         natural log of a         numerical valued
                                                   expression
          LOG(a)        log to base 10 of a      numerical valued
                                                   expression
          SQRT(a)       positive square root     numerical valued
                          of a                     expression
          SIN(a)        sine of a, a in          numerical valued
                          radians                  expression
          MAX(a,b,...)  maximum value of all     any expressions
                          elements of all
                          arguments
^L
                               A-3

          MIN(a,b,...)  minimum value of all     any expressions
                          elements of all
                          arguments
          RAND(a)       next pseudo-random       any expression
                          number, or array
                          of numbers, in a
                          sequence in which
                          a was the last
          DET(a)        determinant of a         square array valued
                                                   expression
          DOT(a,b)      dot product of a         vector-valued expres-
                             and b                 sions of equal
                                                   dimensions
          INV(a)        inverse of a             square array valued
                                                   expression
          POSMAX(a)     row position of maxi-    array valued expression
                          mum element of a
          POSMIN(a)     row position of mini-    array valued expression
                          mum element of a
          SGM(a)        sigma of a (sum of       array valued expression
                          all elements)
          TRC(a)        trace of a (sum of       array valued expression
                          elements on prin-
                          cipal diagonal)
          TRN(a)        transpose of a           array valued expression

     Relations
     ---------
          Symbol   With scalar expressions   With array expressions
          ------   -----------------------   ----------------------
            =      equals                    all corresponding ele-
                                               ments equal
            NE     not equal to              at least one pair of
                                               corr. ele. not equal
            LE     less than or equal to     all corr. ele. less than
                                               or equal
            GE     greater than or equal     all corr. ele. greater
                     to                        than or equal
            LT     less than                 all corr. ele. less than
                                               or equal; at least one
                                               less than
            GT     greater than              all corr. ele. greater
                                               than or equal; at least
                                               one greater than

     [See 11-4 for round-off rules applying to relations]

STATEMENTS

     The following symbols are used in the statement descriptions:

          v1, v2, ...            variables (scalar, vector, or
                                 matrix except as noted)
^L
                               A-4

          r1, r2, ...            relations

          slabel1, slabel2, ...  statement labels

          blabel1, blabel2, ...  block labels

          e1, e2, ...            arithmetic expressions (a meaningful
                                 and conformable combination of
                                 numbers, variables, functions, and
                                 arithmetic operators)

Statements should begin in column 10 of the programming form.  If
continued onto more than one line, the second and aubsequent lines
should begin in column 15.  Columns 73 to 80 must not be used.

Any statement may be given a label--beginning in column 1 of the
form.

     Assignment Statement
     --------------------
          LET v1 = e1

     Sequence Control Statements
     ---------------------------
          GO TO slabel1

          GO TO blabel1        Used only inside block 'blabel';
                               causes skip to end of block.

          IF e1 r1 e2 THEN s1 ELSE s2    where s1 and s2 are any type
                                         of statement except IF or
                                         PERFORM.

                      Either the THEN phrase or the ELSE phrase may
                      be omitted, but one or both must be given.

               Compound conditions may be used:

                      IF e1 r1 e2 AND e3 r2 e4 AND ... THEN s1 ELSE s2

                      IF e1 r1 e2 OR e3 r2 e4 OR ... THEN s1 ELSE s2

               but AND and OR phrases may not be mixed in the same
               statement.                 ---

          STOP

     Iteration Control Statements
     ----------------------------
          A 'block' consists of a sequence of statements preceded by

              blabel1 BLOCK   and followed by   blabel1 END
^L
                          A-5

     A block may be located anywhere in the program; it is executed
         only by a PERFORM statement calling it by name.  Blocks
         may be nested but not overlapped.  A block may contain
         any kind of statent, including PERFORM, except for a
         PERFORM referring to the block itself.

     PERFORM blabel1

     PERFORM blabel1 e1 TIMES    where e1 has integer value.

     PERFORM blabel1 WHILE e1 r1 e2

         Compound conditions may be used:

              WHILE e1 r1 e2 AND e3 r2 e4 AND ...

              WHILE e1 r1 e2 OR e3 r2 e4 OR ...

         but AND and OR phrases may not be mixed in the same
         statement.

     PERFORM blabel1 FOR v1 = e1, e2, ...

                     FOR sv1 = e1 TO e2 BY e3    where sv1 is
                                                  a scalar vari-
                                                  able

                          The order of the TO and BY phrases can
                          be reversed; the BY phrase can be
                          omitted if e3 = 1.   [See 8-3 for more]


     Communication Statements
     ------------------------
          READ v1, v2, ...   [See 5-2 and 9-2 for description]

          WRITE v1, v2, ... , 'title message', ... , /v3, /v4

                    Three types of elements may appear in the list
                    after WRITE:

                       1.  Variable.  Prints:

                              name of scalar and current value;

                              name of vector and current values of
                              components;

                              name of each row vector of matrix and
                              current value of components.

                       2.  Variable preceded by a / . Current values
                              only are printed.
^L
                               A-6

                       3.  A message enclosed in single quotes.  The
                              exact image will appear as a title on
                              the output.  Any characters except the
                              quote may be used in such a message.  A
                              message cannot continue onto a second
                              line on the programming form--a separate
                              item in the same or another WRITE state-
                              ment must be used.

                       [See 5-3 and 8-10] for more formatting details]

          [WRITE ALL  See 4-12 and 8-3 for a description]

          COMMENT   A comment line can be inserted at any time by
                    writing COMMENT in the label field (columns 1-7) of
                    the programming form.  Such a line will appear in
                    the program listing, but has no effect on execution.


     Dimensioning of Vectors and Matrices
     ------------------------------------
          ALLOCATE mv1(e1, e2), vv2(e3), ...  where mv1  is a matrix
                                   variable and vv2 is a vector
                                   variable, and e1, e2 and e3 have
                                   integer values.

              When space is initially allocated to an array, the
              values of all the elements are zero.  If space is
              later changed by another allocation the values of
              those elements common to the old and the new alloca-
              tions are unchanged; the values of new elements are
              zero.  [See also 9-1]

     Tracing Changes in Value During Execution
     -----------------------------------------

          WATCH v1, v2, ...   where v1  and v2 are scalar variables.

              This will cause the system to monitor the values of
              each of the variables listed and print the new value
              each time one of the listed variables is assigned a
              value by a LET or READ statement.  This operation is
              temporary and is automatically discontinued for a
              particular variable after 10 such assignments.  [See 8-3]


DATA

     Data to be read by the execution of the READ statements is
     provided on the same form, after the last statement of the
     program.  The first data line is indicated by writing *DATA
     in columns 1 to 5.  Data items may be entered on this line
     beginning in column 7 and in columns 1 tto 72 of any following
     lines

^L
                               A-7

     Data items are separated by commas.  An item may be either a
     number,  n1,  or an expression of the form  v1 = n1 .  The
     latter form is for checking purposes--it must correspond to the
     variable v1 in the associated READ statement.

     Data items are read in sequence as the program is executed.
     To an array with q elements appearing in a READ statement,
     there must correspond q successive items in the data list.
     If the array is a matrix, the items must be ordered by rows.

---

Other quotes:

From 4-12:

---
     Another statement that can be used for checking purposes is

           WRITE ALL

This will cause the values of all of the variables in the program to be
printed.  There is no WATCH ALL statement.
---

From 5-2:

---
     If READ v occurs in the program and the corresponding entry on the
datalist is w=n, where w is not the same as v, an error message is given.
The value n is assigned to v, no change is made in w, and the program
continues.  Thus the inclusion of items of the type v=n in the data list
provides checks against the accidental omission of data or inclusion of
extra numbers.

     If the total data list is too short, the machine will give an error
message and will give the value 1 for each of the missing numbers.  If the
list is too long the extra entries will be ignored but no error message will
be given.
---

From 5-3:

---
     Numbers are given to 9 significant figures. A number n /= 0 in the range
.0001 <= |n| < 100000 is printed in the usual decimal form, e.g. -327.512736,
0.0243472150 .  Zero is printed simply as 0.  All other numbers appear in the
form mEp with 1 <= |m| < 10, e.g. -2.31562007E+04, 5.00000000E-17 .

     A line of output is divided into six "fields", each 20 characters long.
Each variable name or value occupies one field.  The decimal point of a
number always comes at the seventh position in a field.  Each WRITE statement
starts a new line, but within a given WRITE statement fields are used
consecutively, new lines being started as necessary.  The one exception occurs
when the name of a variable would come at the end of a line and its value
on the next line; in this case, the last field is left blank and the name
starts the next line.

     A field may be purposely skipped by simply omitting a variable name;
for example,

          WRITE ,A,,,/B

will put A and its value in fields 2 and 3, and the value of B in 6,
leaving fields 1, 4 and 5 vacant.  The statement

          WRITE

will cause a whole line to be skipped.
---

From 8-3:

---
     The only restrictions on the use of a subscripted variable are in

          PERFORM k FOR v = ---

and

          WATCH v,...

Here v must be a non-subscripted variable.  Also, WRITE ALL will print only
the values of the non-subscripted variables.
---

From 8-10:

---
     1. A new line is started for each vector and each row of a matrix.

     2. Te values of the elements of a vector or a row of a matrix are
put successively in fields 2 through 6, repeating as necessary.

     3. In field 1 of the first line used for a vector or for a row of
a matrix is put the name of the vector or a symbol for the row of the matrix,
unless these are suppressed by a slash before the array in the WRITE list.
The slash does not change the spacing described in 2 above.

     4. A variable, subscripted or not, appearing in the WRITE list immediately
after an array, starts a new line.

     There is one exception to these rules.  If M is a matrix with only one
column, to save space it will be printed as if it were a vector, that is, in
the form

          M = m11      m21      m31     etc.
---

From 9-1:

---
Vectors as matrices
-------------------
     Consideration of relations involving matrices and vectors can be
simplified by regarding a vector as a 1-column matrix.  This convention is
adopted in CUPL, and so, for example

          ALLOCATE X(7)

and

          ALLOCATE X(7, 1)

have precisely the same meaning.  After either of these allocations the
variables X(2) and X(2, 1) are meaningful and have the same value; X(M, N)
is meaningful only if N has the value 1.
---

From 9-2:

---
     For any matrix M, N(*, J) denotes the 1-column matrix (a vector)
which is the J-th column of M.  Similarly, M(I, *) denotes the 1-row matrix
(not a vector) which is the I-th row of M.  If M is m x n, then M(*, J) and
M(I, *) are m x 1 and 1 x m, and they can be used as matrices of these
sizes in any statement except ALLOCATE.  For example:

     READ M(*, 3)

will read data into the third column of M, leaving the rest of M unchanged.
---

From 9-8:

---
     So much space is needed to compute INV(A) or DET(A) that the size of
A is limited to 40x40 in these expressions.
---

From 11-3:

---
Automatic Integer Round-off
---------------------------
a. The value of a subscript is rounded to the nearest integer.

b. If the round-off involves a change of greater than 10^-9 (approximately)
   an error message is given.
---

From 11-4:

---
Automatic Relative Round-off for x r y
--------------------------------------
a. If both x and y are zero the condition is applied as it stands.

b. If either x or y is not zero:

   (i)   Both x and y are multiplied by 10**n, where n is chosen so that
         the larger of |x * 10**n|, |y * 10**n| lies between .1 and 1.

   (ii)  x * 10**n and y * 10**n are truncated to 14 decimal places

   (iii) The specified condition is interpreted on the resulting numbers.
---

                                APPENDIX F
                       Summary of the CORC Language

     Acceptable Characters
     ---------------------
          Letters: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
          Digits:  0 1 2 3 4 5 6 7 8 9
          Special Characters:  + - / ( ) * $ , . =


     Numbers
     -------
          Normal decimal usage -- + sign may be omitted; decimal point
                    may be omitted from an integer.
          Only 11 significant figures will be considered (8 on Burroughs
                    220)
          Output: 8 significant figures.
          Acceptable range: Absolute values from 10^-308 to 10^+308
                    (10^-49 to 10^+50 on Burroughs 220), and 0.
          Scientific notation: 1.2345 x 10^6 may be written 1.2345*10$(6).

     Variables and Labels
     --------------------
          a. 1 to 8 letters or digits; no blanks or special characters.
          b. First character must be a letter.
          c. Must not be one of the following "reserved" words:

                    ABS     DECREASE   GO         LET   NOTE      STOP
                    AND     ELSE       GTR        LN    OR        THEN
                    ATAN    END        IF         LOG   RAND      TIMES
                    BEGIN   EQUAL      INC        LSS   READ      TITLE
                    BY      EXP        INCREASE   MAX   REPEAT    TO
                    COS     FOR        INT        MIN   SIN       UNTIL
                    DEC     GEQ        LEQ        NEQ   SQRT      WRITE

          d. Statement labels must be unique -- particular label appears only
                    once in label field.
                    Block label used for only one set of BEGIN-END statements.
          e. Variables must be listed in Dictionary. If no initial value
             is given zero is assumed.  [See *1]
          f. One or two subscripts, enclosed in parentheses.  May be
             any expression (including subscripted variables) whose
             value at execution is a positive integer not greater than
             the maximum declared for this variable in Dictionary. [See D-2]
             In Dictionary and Input Data subscripts given as integers in
                    subscript field; parentheses are not used.  [See *1]
^L
     Arithmetic Operators
     --------------------
          + Additions
          - Subtraction
          / Division
          * Multiplication (must be expressed, no implicit multiplication)
          $ Exponentiation; a^b written as a $(b)

     Rules of Precedence
     -------------------
          a. Expressions in parentheses first, from inner to outer.
          b. $ before *, /, + or -
          c. * or / before + or -
          d. Sequence of + and - from left to right

     Functions
     ---------
          Argument any expression, a:
                    ABS(a)   |a|
                    EXP(a)       e^a
                    SIN(a)   sin a, a in radians
                    COS(a)   cos a, a in radians
                    ATAN(a)  arctan a, a in radians
                    INT(a)   [a], greatest integer less than or equal to a

          Arguments two or more expressions, a, b,  ... f:
                    MAX(a, b, ... , f) value equal to greatest of expressions
                       listed
                    MIN(a, b, ... , f) value equal to least of expressions
                       listed
          Argument  any positive expression, b:
                    LN(b) or LOG(b) log b, natural logarithm of b
                                       e
          Argument any non-negative expression, b:
                    SQRT(b)        + b^-2
          Argument a variable, v:
                    RAND(v) next in a sequence of pseudo-random numbers.
                        See Chapter 7

     Relations (used only in IF and REPEAT ... UNTIL statements)
     ---------
          EQL =              NEQ !=
          LSS <              LEQ <=
          GTR >              GEQ >=
^L
     Card Numbers
     ------------
          Strictly increasing sequence of 4 digit numbers, beginning
               with 0010.
          Initially right digit should be zero on all cards to leave
               room for later additions.

     Program Arrangement [See *1]
     -------------------
          Arrange and number programming forms in the following order:
                    1. Preliminary Description
                    2. Dictionary [See 2-10]
                         List all variables to be used
                         Specify initial values if not zero
                         Specify maximum value for each subscript.
                    3. Program Statements
                         One statement per line; for continuation place
                              C in column 21 of following line and begin
                              in column 42.  Do not break in middle of
                              variable, label, or number.
                         THEN, ELSE, OR and AND phrases of IF statement
                              on following lines, beginning in col 42,
                              but C in col 21 not required.
                    4. Input Data
                         Variables, Subscripts and values of data to
                              be called by READ statements in program.

     CORC Statements
     ---------------
     The following symbols are used in the statement descriptions:

          v, w, x   variables
          r, s, t   relations
          a, c      statement labels
          b         block label
          e, f, g, h, j, k    arithmetic expressions (any meaningful
                                 combination of numbers, variables,
                                 functions and arithmetic operators)

     Computation Statements
     ----------------------
          LET v = e
          INCREASE v BY e    or   INC v BY e
          DECREASE v BY e    or   DEC v BY e
^L
     Sequence Control Statements
     ---------------------------
          GO TO a             Statement with label a to be executed next.
          GO TO b             Used only inside block b; causes skip to
                              END of block b.

          IF e r f            Go to statement a if condition e r f is
             THEN GO TO a     satisfied; otherwise go to statement c.
             THEN GO TO c

          IF e r f            Go to statement a if all of the conditions
             AND g s h        listed are satisfied; otherwise go to
             ...              statement c.
             THEN GO TO a
             THEN GO TO c

          IF e r f            Go to statement a if all of the conditions
             OR g s h         listed is satisfied; otherwise go to state-
             ...              ment c.
             THEN GO TO a          AND and OR phrases cannot be mixed in
             THEN GO TO c          the same IF statement.

          STOP                Last statement in execution of program; not
                              necessarily written last on Program State-
                              ment sheet.

     Iteration Control Statements
     ----------------------------
          b BEGIN  Define limits of a block; b in label field, BEGIN-END.
          b END    statement field.
               1. Block may be entered (executed) only by REPEAT statement.
               2. Block may be located anywhere in program.
               3. Blocks may be nested, but not overlapped.
               4. Block b may contain any type of CORC statement, in-
                  cluding REPEAT, but not REPEAT b, ... , a REPEAT
                  Statement referring to itself.
          REPEAT b e TIMES     Value of e a non-negative integer.
          REPEAT b UNTIL e r f Continue repetition of block b until
                               condition e r f is satisfied.
          REPEAT b UNTIL e r f AND g s h AND ... Continue repetition
                               of block b until all conditions listed
                               are satisfied.  (Not available on Burroughs
                               220)
          REPEAT b UNTIL e r f OR g s h OR ... Continue repetition of
                               block b until any one of the conditions
                               listed is satisfied.  (Not available on
                               Burroughs 220)
^L
          REPEAT b FOR   v = e, f, g, ..., ..., (h, j, k), ... Repeat
                               block b once for each expression on list,
                               with value of expression assigned to
                               variable v.  Three expressions on list
                               enclosed in parentheses mean from h to k
                               in steps of k. [See 5-8]

     Communication Statements
     ------------------------
          READ v, w, x, ...    Read an Input Data card for each variable
                               on list; variables on cards read must
                               agree with variables on list.

          WRITE v, w, x, ...   Print variable and current value, three
                               to a line.  Each WRITE statement starts a
                               new line.

          TITLE message        Print "message" in computational results
                               when this statement is encountered in
                               execution of program.

          NOTE message         "Message" will appear in copy of program
                               only, not in execution.  Used for program
                               notes only.

---

[*1] This implementation of CORC does not support or allow a
Dictionary section.  Instead, variable initializations must be done via
CUPL-style DATA and ALLOCATE statements.

[*2] Ignore this section.  Program statements are free-format, with
continuations not supported (though the example program
test/title.corc shows the syntax, it will break cupl).  Data for read
statements is accepted in CUPL format following the keyword *DATA.

For completeness, however, the Dictionary feature is documented here.

From 2-10:

---
Dictionary of Variables
-----------------------
     In addition to a set of statements CORC requires a pro-
gram to contain another part known as the Dictionary.  The Diction-
ary of a program is merely a list of all the variables used in
the program. along with, if desired, the initial assigned values of
the variables.  If no initial value is specified the computer
assigns the initial value zero.
---

From 2-11:

---
     In the above example the Dictionary might look like this:

     A     1
     B    -1
     C    -6
     ROOT
     X1
     X2
---

The above example is the simplecorc.corc program.  The CORC
Dictionary is equivalent to a CUPL DATA section, but also allowed the
programmer to dimension array variables.  The example form on 2-12
makes it clear that the Dictionary was distinguished from the program
proper by being in a different lower range of card line
numbers.

From 5-8:

---
Three expressions on a list, enclosed in parentheses, are in-
terpreted in the following way:
     1. The first expression gives the initial value for the vari-
        able.
     2. The second expression gives the difference between con-
        secutive values.
     3. The third expression indicates where to stop -- the final
        value for the variable is less than or equal to the value
        of the third expression.

[... examples omitted ...]

    More than one such "triple" may be used on a list, and
"triples" may be intermixed with separate expressions;
---

From 6-5:

---
   A particular variable either has no subscripts, one sub-
script, or two subscripts and this use must be consistent through-
out a program.  A variable cannot appear as X(I) in one statement
and X(I, J) or just X in another statement of the same program.
The nature of a variable (whether it is to be subscripted or not)
must be indicated when the variable is listed in the CORC Diction-
ary.  This is done by giving the maximum value of any subscripts
that will be used in columns 21-25 of the Dictionary form.  If no
subscripts will be used these columns will be left blank in the
line for that variable.  If one subscript is to be used, the maxi-
mum value that that subscript will take on anywhere in the pro-
gram must be given in columns 21-23; columns 24-25 are left blank.
(Note that this is the maximum value of the subscript, and not
the maximum value of the variable.)  If two subscripts are to be
used the maximum value of the first is given in columns 21-23
and the maximum value of the second in columns 24-25.  For ex-
ample, if the Dicitonary [sic] looks like the following:

          SCALAR
          VECTOR    45
          MATRIX   100     2
          ARRAY      3    32

then SCALAR is a simple variable that will not have any sub-
scripts anywhere in the program.  VECTOR will have one subscript
everwhere it appears in the program [...] MATRIX [...] will
appear each time with two subscripts [...] ARRAY will also
always have two subscripts; [...]
---

From D-2:

---
Automatic Integer Round-off
---------------------------
a. The value of a subscript is rounded off to the nearest
   integer.
b. If the round-off involves a change of greater than
   10^-9 (approximately;   the number is subject to some
   variation) an error message is given.
---

From D-3:

---
Automatic Relative Round-off for x r y
--------------------------------------
a. If both x and y are zero the condition is applied as it stands.
b. If either x or y is not zero:
   (i) Both x and y are multiplied by MAX(|x|, |y|);
   (ii) The results are rounded off to the nearest
        integer if this involves a change of less
        than 10^-9 (10^-7 for the Burroughs 220), but
        not otherwise;
   (iii) The specified condition is interpreted for
        the resulting numbers.
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