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         Volume 2 Number 3        48/39                 March 1977

                     Newsletter of the SR-52 Users Club
                                published at
                           9459 Taylorsville Road
                              Dayton, OH 45424
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Random Numbers
     Gerald Donnely (203) brings up the topic of random numbers, and
how best to generate them for particular applications.  I should note
at the onset that mathematicians are not even in agreement on a defini-
tion, let alone how best to generate them!  However, for games applica-
tions with the SR-52/56 machines perhaps we can agree on the following
admittedly fuzzy statements:  The "best" random number generator will
select numbers one at a time from a given set, producing a sequence
whose ordering and distribution characteristics most closely resemble
those produced by successive spins of an appropriately marked, unbiased
roulette wheel.  There are two general mechanization approaches: what
I will term open, and closed.  Open requires manual intervention via
program interrupt; closed is self-contained, and produces sequences
which are predetermined by the value of an initializing "seed" number.
The open approach comes the closest to the stated concept of best, but
imposes a chore on the user; the closed approach produces predictable
sequences, but is automatic.  The closed approach is used most often,
and is generally satisfactory, as most users find it either distastefully
ungentlepersonly or too much bother (or both) to determine in advance
each predictable sequence.  But perhaps the open approach deserves some
attention, since it is easily mechanized in-relatively few steps.
     If random numbers in, say, the range 1-6 do not need to be hidden
from the user, something along the lines of:  *LBL A 5 = 1 = 4 = 6 =
3 = 2 = GTO A will work quite well:  press A, then HLT.  The longer the
time between pressing A and HLT, the more "random" the selection is
likely to be.  If the numbers need to be hidden:  *LBL A RCL 05 *EXC 03
*EXC 06 *EXC 01 *EXC 04 *EXC 02 STO 05 *pi sin *ifzro A 0 HLT might
do, and is run with prestored values in Reg 01-06 by repeating:  press
A and switch from radian to degrees at an arbitrary time.  A main
program would use the contents of any one of Reg 01-06.
     The closed approach poses two problems:  how to keep generated
numbers within a desired range, and how to get a "good" distribution,
all in a minimum number of steps.  The severity of the range problem
depends on the required range.  If the only restriction is that genera-
ted numbers fall between 0 and 1, the range problem is trivial.  But
a requirement, say, to produce a random ordering of the first ten primes
might take such a chunk of memory space and/or execution time as to
defy practical implementation.  But let's look at something in between,
As a "working" number, pi is a handy 13-digit, middle-magnitude, posi-
tive real whose digit values form a "randomly" distributed string of
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  The SR-52 Users Club is a non-profit loosely organized group of SR-52/56 owners/users
  who wish to get more out of their machines by exchanging ideas.  Activity centers
  on a monthly newsletter, 52-NOTES edited and published by Richard C Vanderburgh
  in Dayton, Ohio.  The SR-52 Users Club is neither sponsored nor officially sanctioned
  by Texas Instruments, Incorporated.  Membership is open to any interested person:
  $6.00 includes six future issues of 52-NOTES; back issues start June 1976 @ $1.00 ea.

8 of the 10 numerals, and every closed random number generator I've
seen for the personal programmables uses it in one way or another.  A
general approach is to combine pi with a seed by some succession of
arithmetic operations, replace the seed with the fractional part of
the result, and convert the most significant digit(s) of this result
into an integer which may need to be scaled, depending upon range
requirements.  Each successive generation combines pi with the previously
generated seed.  Although I've seen square roots and fifth powers
applied, a simple multiplication of pi by the seed seems to be quite
satisfactory.  As an example, let's look at Routine D' of Alan Charbon-
neau's Yahtzee program (V2N2p5).  The requirement is to randomly select
one of the numerals 1-6 each time the routine is called.  The fractional
part of the seed-times-pi product is saved, and then multiplied by
the scaling factors 5.5 which guarantees a real roundable to an integer
in the 0-5 range.  The + 1 offsets the range to 1-6.  Although this
scaling method works (one of my revisions to Alan's program), Steve
Marum (188) points out that there is a bias favoring the numerals 2-6.
His revision:  022:  RCL 98 X 6 + .5 = ...  removes the bias (caution:
when making this substitution into Alan's program, either include a
neutral spacer, or change all absolute addresses and reposition the
unrelocatable code at steps 208-223).
     Rather than go on with other requirements and approaches now, I'll
wait for input from the membership.  Send me your prize random number
generators, hard-to-meet requirements, pedagogical arguments, etc and
we'll resume discussion in a future issue of 52-NOTES.

A Sum of the Digits Challenge
     Jared Weinberger (221) has been experimenting with various
approaches to summing the digits of a number.  For the SR-52 the seq-
uence:  *LBL A (EE INV *fix EE 00 - INV *D.MS INV *D.MS *fix 0 EE SUM
69) INV *ifzro A *EXC 69 *rtn is Jared's fastest, while:  *LBL B X 1
SUM 69 - EE div EE 00 = INV *ifzro B *EXC 69 *rtn is shorter but slower.
Both require a one-time zeroing of Reg 69.  Upon call, the displayed
number's digits are summed, and the sum returned.  For the SR-56,
advantage can be taken of the Int function, and the routine starting
at 00:  (EE INV *fix EE 00 - *Int SUM 1) *x=t 16 RST *EXC 1 *rtn is
both shorter and faster than either of the SR-52 ones.  The routine:
X 1 SUM 1 - EE div EE 00 = *x=t 15 RST *EXC 1 *rtn is one step shorter,
but takes longer to run.  To run these SR-56 routines, zero Reg 1 once
along with the t register, then RST R/S with the number whose digits
are to be summed in the display.  Can anyone beat 21 SR-52 steps or 18
SR-56 steps?

PPX-52 Gems and Rejects (52)
     Stan Wagon (456) notes that program #390016:  Extra Precision
Factorials by one Greg Evans "... is interesting and might prove useful
to anyone contemplating using double precision in other contexts."
     E. S. (Mack) Maloney (246) has a great circle navigation program
rejected by TI as a duplicate, but which calculates an initial course
from the point of departure (which #949025 reportedly does not).  Send
Mack a SASE and 25¢ if you would like a copy of his program.

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                              52-NOTES V2N3p2

Book Review
     Statistics for the SR-52: Keyboard Data, 188 pages $10.00;
Statistics for the SR-52: Card Data, 100 pages $11.00; and Statistics
for the SR-56, 79 pages $9.30; all by C Donaldson Ellis, published 1977
by Wilmardon Publishing Box 461 Storrs, CT 06268.
     All three volumes are printed in a pica typewriter font, one side
of 8½ x 11 paper, with direct copy of PC-100 printer listings opposite
manually generated mnemonics for program listings.  Each program begins
with a brief description of what it does (assuming user familiarity
with statistics usage), along with special requirements and limitations.
User instrctions are contained in a step-numbered  Procedure section.
This is followed by a table of register contents, and the program list-
ing.  Applicable formulas and algorithms are noted as being found in
one or more of five references, but are not printed in the text; there
are no sample problems.  All programs are written for use with the
PC-100, with notes covering required modifications for SR-52/56 use
only.
     The SR-52 keyboard manual covers 54 statistics programs, some
requiring more than 224 steps.  Input data are entered as required from
the keyboard.  The card manual arranges half of the keyboard programs
to allow for data storage and retrieval via mag cards.  Data packing and
unpacking routines are given, but actual card storage is by program
step only (no mention is made of direct storage to/from program regis-
ters.  Register tables show data assignments to Reg 98, 99, 00, ... 19.
I will not comment on the degree to which programs operate satisfac-
torily or have been optimized, except to note that a 2 X 2/ 3 X 3
matrix inversion program requires 445 steps, and one titled "Four-way
ANOVA--Unweighted Means" requires 1736 steps (8 cards).
     The SR-56 manual follows along the same lines, covering 40 programs,
some of which require more than 100 steps.  The longest program is
"Mixed ANOVA 1 Within and 1 Between" which requires 242 steps (3
programs to key in).
     It would appear that these manuals would be the most useful to
statisticians who have access to SR-52/56 machines, but who don't know
how (or don't care) to program them.  Users are invited to communicate
with the author via the publisher.

A Subtle Danger in Using "iferr" as a Flag (52)
     Barbara Osofsky (420) notes a potential danger in using an undefined
letter function to set an error condition flag in programs where
program storage registers are used for data.  In my version of her 5X5
program (V2N2p3,4) if any of the first 3 data entries begin with the
sequence: 1346 or contain this sequence in the following patterns:
XX1346, XXXX1346, or XXXXXX1346, then LBL C becomes defined (see V1N1p4)
and the user call to C would cause undesired execution of the code
(or data) following the LBL C.  While the probability is small that
LBL C would be unintentionally defined, the potential is there.  As an
example of a worst (albeit improbable) case, if the first 2 elements
of a 5X5 matrix are: 8.594011346 and -1.234567895D64 respectively,
when C is pressed, there is total wipeout! (see V1N2p2).  As it turns
out, Barbara wat able to accomplish the conditional branch with a real
flag in a recent version of her program which I may publish later.  In
the meantime, users of the V2N2 5X5 program who expect to input matrix
elements containing more than 3 digits may wish to make the following

                              52-NOTES V2N3p3

changes:  re-write, beginning at step 192:  STO 69 + = HLT.  (The
HLT at step 199 may be either left alone, or deleted).  Re-write step
6 of the user instructions to read:  6.  Key 9n, press RUN, then E; jth
column printed, j=2,3,4,5, see 1 displayed; repeat for n=6,7,8,9.
Re-write note 3 to read:  3.  At step 6, following the RUN, 5 is
flashed.
     It should he noted that any time program registers are used for
data the possibility of unintentional labeling exists.  However, if
such registers are chosen at the high end (... 95, 96, 97) duplicate
labels pose no problem since the machine always searches for labels
starting at step 000.

A Utility Program for Fractured Digits (52)
     In modern programming terminology, a utility program is one which
aids the programmer in developing other programs.  Routine A in V1N5p5
(top) is a short special purpose utility routine that helps the user
synthesize look-up tables which become parts of other programs.  As
noted, once a main program has been written, Routine A is no longer
needed, and it can be shelved until such time as it might be used to
help in writing another program.  A while back I attempted to arrive
at the "best" display format for a game program I was working on-by
trying out various sequences of fractured digits mixed in with numer-
ical results.  I soon realized that it would help to have a fractured
digits synthesizer program that could produce specified patterns in
the SR-52 display, so they could be seen and judged as they actually
appear.  In the following program, the user specifies the character he
wishes to appear at each of the 14 display positions (within the con-
straints described in V1N2).  0 produces the numeral 8 (which arbitrar-
ily represents any desired numeral), 2 produces the " symbol, 3 pro-
duces a blank, 4 a ' symbol, 5 a ° symbol, and 6 a - symbol.  Inci-
dently, a 6 for position 11 followed by 3s for positions 12 and 13 will
produce a - at position 11 and blanks at positions 12 and 13 (contrary
to a statement in V1N2p5).  As each position is processed, the
number left in the display indicates the next position to be specified.

SR-52 Utility Program:  Display Variations Synthesizer        Ed

User Instructions:
     1.  Key position 0 code, press A; see 1 displayed
     2.  Key ith position code, press RUN; see i+1 displayed
         Repeat for i=1,2,...13
     3.  Press =, see synthesized display
     4.  To synthesize a new display, press CLR and go to step 1

Program Listing:

000:  *LBL B = SUM 98 *LBL C 1 SUM 18 RCL 18 INV EE HLT *rtn *LBL A INV
022:  *stflg 0 INV *stflg 1 - 3 = *ifzro 036 *stflg 0 1.11 STO 98
043:  RCL 92 STO 19 1 STO 18 HLT div 1000 B div 1 EE 4 B div 1 EE 5
069:  B div 1 EE 6 B div 1 EE 7 B div 1 EE 8 B div 1 EE 9 B div 1 EE 10
095:  B div 1 EE 11 B = C - 3 = *ifzro 113 *stflg 1 C div 1 EE 12 B X
121:  10 = STO 18 10 yx RCL 18 = EE *fix0 *PROD 98 1 EE 88 INV *ifflg 1
147:  151 *1/x INV *ifflg 0 158 +/- *PROD 19 RCL 98 + STO 60 RCL 19
171:  INV EE *fix 0 HLT 0 *ifpos *2' *2' *2' *2' *2' A

                            - - - - - - - -

                              52-NOTES V2N3p4

An AOS Advantage for a Problem in Logic (52)
     As most of us know, AOS vs RPN arguments rarely end in conclusive
victories for either side.  But here is one application where AOS does
appear to have a clear advantage.  Logician Stan Wagon (456) found
that the SR-52's nested parentheses could be made to order hierarchies
in certain types of logical statements.  In a branch of mathematical
logic sometimes referred to as propositional calculus, 2-state variables
which assume values of either true or false are operated on by the 5
logical operators:  NOT, AND, OR, IF...THEN, and IF AND ONLY IF.  All
but NOT are "connectives", that is, they link two variables or paired
parenthetically grouped variables.  A string of variables, logical
operators, and parentheses forms a logical statement or sentence.  For
each possible configuration of variable states, the entire statement
is either true or false, and a tabulation of all such configurations is
known as a Truth Table.  (See Chapter 1 of the Boolean Algebra and
Switching Circuits Schaum's Outline if you are new to the subject).
Given the elementary truth tables:
                       |       |       | IF... | IF AND  |
       NOT             | AND   |  OR   | THEN  | ONLY IF |
   |A| ¬A |      A | B | A & B | A ∨ B | A→B   | A↔B     |
   |T|  F |      T | T |   T   |   T   |  T    |  T      |
   |F|  T |      T | F |   F   |   T   |  F    |  F      |
                 F | T |   F   |   T   |  T    |  F      |
                 F | F |   F   |   F   |  T    |  T      |

Stan's Truth Tables program that follows accepts a statement of n
variables (0 < n < 10) and up to 27 characters in length, and prints
(or displays) the statement's truth or falsity for all 2n possible ways
to assign truth values to the n variables.  Each statement as written
in LRN mode as the body of Routine E', and must be contained within a
parenthesis pair.  4 of the 5 logical operators are assigned to letter-
functions as follows:  A'=OR, B'=NOT, C'=IF...THEN, D'=IF AND ONLY IF;
the "times" operator X=AND.  Parentheses should be used extravagantly
(when in doubt, use them).  The 5 letter functions A, B, ... E
correspond to the first 5 variables.  For up to 4 more: RCL 06, RCL 07,
RCL 08, RCL 09 would be written in Routine E' to represent the additional
variables.  In order to adopt the output convention that 0=false and
1=true, Stan needed a way to vary the number of displayed digits, and
worked out a dynamic-code-change method.  A skeleton block of 8 instruc-
tions (steps 184-191) held permanently in Reg 93 is modified during
program execution to supply the proper value for n in "fix n", then
put into Reg 91, which supplies steps 168-175.  A logical statement (S)
that might be written out as:  ((a&b)∨(c→¬b))→(a∨b) would be written
in Routine E' as:  *LBL *E' (((A X B) *A' (C *C' (*B' B))) *C' (A *A' B))
*rtn.
     Each displayed line of the truth table shows the truth or falsity
of the statement as an integer 1 or 0.  Variable truths or falsities
are shown to the right of the decimal.  In the above example, the
tenths place indicates the truth or falsity of variable a, the hundredths
place b, and the thousandths c.  The resulting truth tables are dis-
played as: 1.111, 1.110, 1.101, 1.100, 1.011, 1.010, 0.001, and 0.000,
and translate to:

                              52-NOTES V2N3p5