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OS\BITS/T\OSB23
This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.
Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.
| Tape/disk: | Home » CEEFAX disks » telesoftware2.adl |
| Filename: | OS\BITS/T\OSB23 |
| Read OK: | ✔ |
| File size: | 2280 bytes |
| Load address: | 0000 |
| Exec address: | 0000 |
File contents
OSBITS - An Exploration of the BBC Micro at Machine Level
By Programmer
..........................................................
Part 23: Floating Point Arithmetic II
In this module we don't have much in the way of text, but we
have another program using floating point machine code
arithmetic. This time, having achieved addition and
subtraction in our experimentation, we go on to
multiplication and division.
The general rule for multiplication is that you multiply the
mantissae and add the exponents while for division you
divide the mantissae and subtract the exponents.
Thus, if A and B are the mantissae and C and D are exponents
(such that, for example, 12.34 could be represented as
1.234e1 where 1.234 is the mantissa and 1 is the exponent -
the power of 10, in this case, needed to multiply the number
to its actual value) we have the following:
AeC * BeD = (A*B)e(C+D)
AeC / DeD = (A/B)e(C-D)
The program in this module works on this basis but it does
throw up a few problems. I'll go through the program
section by section and briefly discuss the problems I came
across in writing it.
As with the last module, this program is divided into
subroutines that carry out the main sections of the work.
One significant difference is that, this time, I have used
CALL with parameters as a means of getting the floating
point numbers into the program. This means that the code is
only assembled once in each RUN with a REPEAT UNTIL FALSE
loop letting you enter new values and seeing the results
each time. This is an alternative to using the FNEQUF
pseudo-opt in the assembler. You may like to modify the
last module to work in this way.
Having been transferred, using information in the parameter
block at &600 that BASIC sets up for CALL with parameters,
the numbers input are stored securely, since they will be
needed twice, once for multiplication and once for division.
Note that the initial transfer needs some zero page
workspace because of the indirect addressing used. The
block at &600 looks like this:
&600 1 byte - number of parameters
&601 2 bytes - parameter address
&603 1 byte - parameter type
&604 2 bytes - next parameter address .... and so on
The parameter type for a floating point variable is 5.
Others are 0 for a byte (i.e. an ?X), 4 for an integer, 128
for a $X string in memory (terminated by a CR) and 129 for a
string variable (X$). In the last case (see module 7) the
parameter address is a pointer to a string information
block.
The numbers are then transferred into the workspaces for the
calculation by a subroutine 'transfer_in'. The
multiplication is carried out and then the result is
transferred out into 'result_mult'. This process is
repeated for the division. At the end the results of the
machine code calculations are compared with the result using
BBC BASIC. I will admit here and now that my routines are a
little innacurate but I will also admit to this being of an
experimental nature and I was trying for clarity.
As we go down the listing of the assembly program the next
thing we meet is a block of workspace. I have increased the
workspace for the calculations to seven bytes from five
because I need seven for division. I have also triple named
'fpws_1' and 'fpws_2' for their multiplication and division
roles. This is a technique you can use if you want to be
clear about the function of a piece of memory although its
use changes during the program. There is no penalty for
multi-naming, apart from making the assembly source code a
bit longer. Your final program code, which does not use the
labels, is not affected.
So let's look at the multiplication first.
To avoid problems I have trapped zeros at the start of the
multiplying routines. Obviously a zero entered as either
number will give a zero result when they are multiplied
together. There is a subroutine at 'make_zero' which makes
all five bytes of 'fpws_1' zero. Note that if you branch or
jump to a subroutine, rather than JSRing to it, you will
automatically RTS at the end of it, in this case that takes
us out of the 'fp_mult' subroutine completely.
If neither number is zero we move on. The sign bit is
transferred to another byte and the top bit of the mantissa
corrected exactly as in the last module.
The multiplying denormalisation routine rotates each
mantissa right by two bytes. With a four byte mantissa this
effectively gives us two two byte numbers which, when
multiplied together, produce a four byte number. This four
byte number then fills the mantissa work space again.
You might think, as I originally did, that when rotating the
mantissae you should alter the exponents to compensate.
This produces incorrect results and, after much thinking and
drawing of diagrams I came to the following conclusion. We
are do not have to change the exponents because the
mantissae are, in fact, just less that 1 in size. Now if
you multiply 0.9 by 0.9 you get 0.81 and multiplying two
mantissae, each with its top bit set, is just like this. We
get a similar situation with division.
'mult_denormalise' also calculates the initial new exponent.
Remember that BBC BASIC adds &80 to the exponents in its
floating point format. If on adding the exponents we change
the 2's complement sign of the result then the result ha
overflowed and is out of range. This might happen if, say,
we added 120 to 20. 140 is actually a negative number in
2's complement and so we have wrapped around the byte to
give an incorrect result. We can trap this by checking the
overflow flag. This flag will detect an anomalous overflow
and so we can use it to branch to an error routine. With
multiplication, a result that has overflowed too much
negatively means that the number is in fact too small for us
to represent it. This is not treated as an error, but
instead means that we must set the result to zero. For a
value that is too big I have used Error 20, but given it
slightly different messages to BASIC as you can see in the
listing.
I multiplied the two numbers using a modified version of the
multi-byte multiplication routine from module 13. The main
difference is that the floating point mantissa is stored
back to front compared to the integers, with the least
significant byte in the lowest memory location. Note that,
as the partial product workspace does not use the exponent,
it is only four bytes long not five.
After multiplying we renormalise the mantissa by rotating it
left until the top bit is set. By converting to two two
byte numbers we cannot have an overflow here so we only have
to check for a number that is too small. A zero result is
trapped here because it would otherwise send the routine
into an endless loop.
Finally the sign bit is replaced. The sign business is a
great deal easier with multiplcation and division that it
was with addition and subtraction. If you EOR the two sign
bits together you get the result; simple as that.
Where division is concerned much of the procedure is the
same. Here a zero as the second number is a fatal error
since we cannot divide by zero. A zero for first number
just leads to a zero result.
Denormalisation for division, in my implementation, involves
rotating the divisor right by two bytes. This effectively
means that we have a four byte number divided by a two byte
one, although the two byte one has two other bytes beyond
the 'binary point' and the division routine therefore works
over 48 bits.
In calculating the initial exponent of the result I found
that to compensate for the rotation, and because both our
numbers were just under 1 in size again, I had to increase
the size of the exponent by 32 (&20) which is done after the
second exponent has been subtracted from the first. [I wish
I understood these 'fudge' factors more clearly, then I
could explain them more clearly.]
Just as the exponents could overflow during multiplication,
so they can during division, and that is trapped at the end
of the 'div_denormalise' routine.
The division subroutine is a straightforward modification of
the multi-byte integer division routine in module 15. In
this case we calculate over 48 bits instead of 32, and the
workspace is back to front because of the way the floating
point mantissae are stored.
Renormalisation and fixing the sign bit operate exactly as
with multiplication, using the same subroutines.
These floating point routines are by no means simple. To
make them more accurate would make them more obscure, but
the principles behind them are straightforward enough.
Maybe you'd like to try making that Mandelbrot program from
module 21 work using floating point?
While you're thinking about that I will move on to sideways
ROMs, a subject very specific to the BBC Micro, in the next
module.
00000000 4f 53 42 49 54 53 20 2d 20 41 6e 20 45 78 70 6c |OSBITS - An Expl| 00000010 6f 72 61 74 69 6f 6e 20 6f 66 20 74 68 65 20 42 |oration of the B| 00000020 42 43 20 4d 69 63 72 6f 20 61 74 20 4d 61 63 68 |BC Micro at Mach| 00000030 69 6e 65 20 4c 65 76 65 6c 0d 0d 42 79 20 50 72 |ine Level..By Pr| 00000040 6f 67 72 61 6d 6d 65 72 0d 0d 2e 2e 2e 2e 2e 2e |ogrammer........| 00000050 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e |................| * 00000080 2e 2e 2e 2e 0d 0d 0d 50 61 72 74 20 32 33 3a 20 |.......Part 23: | 00000090 46 6c 6f 61 74 69 6e 67 20 50 6f 69 6e 74 20 41 |Floating Point A| 000000a0 72 69 74 68 6d 65 74 69 63 20 49 49 0d 0d 0d 49 |rithmetic II...I| 000000b0 6e 20 74 68 69 73 20 6d 6f 64 75 6c 65 20 77 65 |n this module we| 000000c0 20 64 6f 6e 27 74 20 68 61 76 65 20 6d 75 63 68 | don't have much| 000000d0 20 69 6e 20 74 68 65 20 77 61 79 20 6f 66 20 74 | in the way of t| 000000e0 65 78 74 2c 20 62 75 74 20 77 65 0d 68 61 76 65 |ext, but we.have| 000000f0 20 61 6e 6f 74 68 65 72 20 70 72 6f 67 72 61 6d | another program| 00000100 20 75 73 69 6e 67 20 66 6c 6f 61 74 69 6e 67 20 | using floating | 00000110 70 6f 69 6e 74 20 6d 61 63 68 69 6e 65 20 63 6f |point machine co| 00000120 64 65 0d 61 72 69 74 68 6d 65 74 69 63 2e 20 20 |de.arithmetic. | 00000130 54 68 69 73 20 74 69 6d 65 2c 20 68 61 76 69 6e |This time, havin| 00000140 67 20 61 63 68 69 65 76 65 64 20 61 64 64 69 74 |g achieved addit| 00000150 69 6f 6e 20 61 6e 64 0d 73 75 62 74 72 61 63 74 |ion and.subtract| 00000160 69 6f 6e 20 69 6e 20 6f 75 72 20 65 78 70 65 72 |ion in our exper| 00000170 69 6d 65 6e 74 61 74 69 6f 6e 2c 20 77 65 20 67 |imentation, we g| 00000180 6f 20 6f 6e 20 74 6f 0d 6d 75 6c 74 69 70 6c 69 |o on to.multipli| 00000190 63 61 74 69 6f 6e 20 61 6e 64 20 64 69 76 69 73 |cation and divis| 000001a0 69 6f 6e 2e 0d 0d 54 68 65 20 67 65 6e 65 72 61 |ion...The genera| 000001b0 6c 20 72 75 6c 65 20 66 6f 72 20 6d 75 6c 74 69 |l rule for multi| 000001c0 70 6c 69 63 61 74 69 6f 6e 20 69 73 20 74 68 61 |plication is tha| 000001d0 74 20 79 6f 75 20 6d 75 6c 74 69 70 6c 79 20 74 |t you multiply t| 000001e0 68 65 0d 6d 61 6e 74 69 73 73 61 65 20 61 6e 64 |he.mantissae and| 000001f0 20 61 64 64 20 74 68 65 20 65 78 70 6f 6e 65 6e | add the exponen| 00000200 74 73 20 77 68 69 6c 65 20 66 6f 72 20 64 69 76 |ts while for div| 00000210 69 73 69 6f 6e 20 79 6f 75 0d 64 69 76 69 64 65 |ision you.divide| 00000220 20 74 68 65 20 6d 61 6e 74 69 73 73 61 65 20 61 | the mantissae a| 00000230 6e 64 20 73 75 62 74 72 61 63 74 20 74 68 65 20 |nd subtract the | 00000240 65 78 70 6f 6e 65 6e 74 73 2e 0d 0d 54 68 75 73 |exponents...Thus| 00000250 2c 20 69 66 20 41 20 61 6e 64 20 42 20 61 72 65 |, if A and B are| 00000260 20 74 68 65 20 6d 61 6e 74 69 73 73 61 65 20 61 | the mantissae a| 00000270 6e 64 20 43 20 61 6e 64 20 44 20 61 72 65 20 65 |nd C and D are e| 00000280 78 70 6f 6e 65 6e 74 73 0d 28 73 75 63 68 20 74 |xponents.(such t| 00000290 68 61 74 2c 20 66 6f 72 20 65 78 61 6d 70 6c 65 |hat, for example| 000002a0 2c 20 31 32 2e 33 34 20 63 6f 75 6c 64 20 62 65 |, 12.34 could be| 000002b0 20 72 65 70 72 65 73 65 6e 74 65 64 20 61 73 0d | represented as.| 000002c0 31 2e 32 33 34 65 31 20 77 68 65 72 65 20 31 2e |1.234e1 where 1.| 000002d0 32 33 34 20 69 73 20 74 68 65 20 6d 61 6e 74 69 |234 is the manti| 000002e0 73 73 61 20 61 6e 64 20 31 20 69 73 20 74 68 65 |ssa and 1 is the| 000002f0 20 65 78 70 6f 6e 65 6e 74 20 2d 0d 74 68 65 20 | exponent -.the | 00000300 70 6f 77 65 72 20 6f 66 20 31 30 2c 20 69 6e 20 |power of 10, in | 00000310 74 68 69 73 20 63 61 73 65 2c 20 6e 65 65 64 65 |this case, neede| 00000320 64 20 74 6f 20 6d 75 6c 74 69 70 6c 79 20 74 68 |d to multiply th| 00000330 65 20 6e 75 6d 62 65 72 0d 74 6f 20 69 74 73 20 |e number.to its | 00000340 61 63 74 75 61 6c 20 76 61 6c 75 65 29 20 77 65 |actual value) we| 00000350 20 68 61 76 65 20 74 68 65 20 66 6f 6c 6c 6f 77 | have the follow| 00000360 69 6e 67 3a 0d 0d 20 20 20 20 20 20 20 20 20 20 |ing:.. | 00000370 20 20 20 20 41 65 43 20 2a 20 42 65 44 20 3d 20 | AeC * BeD = | 00000380 28 41 2a 42 29 65 28 43 2b 44 29 0d 20 20 20 20 |(A*B)e(C+D). | 00000390 20 20 20 20 20 20 20 20 20 20 41 65 43 20 2f 20 | AeC / | 000003a0 44 65 44 20 3d 20 28 41 2f 42 29 65 28 43 2d 44 |DeD = (A/B)e(C-D| 000003b0 29 0d 0d 54 68 65 20 70 72 6f 67 72 61 6d 20 69 |)..The program i| 000003c0 6e 20 74 68 69 73 20 6d 6f 64 75 6c 65 20 77 6f |n this module wo| 000003d0 72 6b 73 20 6f 6e 20 74 68 69 73 20 62 61 73 69 |rks on this basi| 000003e0 73 20 62 75 74 20 69 74 20 64 6f 65 73 0d 74 68 |s but it does.th| 000003f0 72 6f 77 20 75 70 20 61 20 66 65 77 20 70 72 6f |row up a few pro| 00000400 62 6c 65 6d 73 2e 20 20 49 27 6c 6c 20 67 6f 20 |blems. I'll go | 00000410 74 68 72 6f 75 67 68 20 74 68 65 20 70 72 6f 67 |through the prog| 00000420 72 61 6d 0d 73 65 63 74 69 6f 6e 20 62 79 20 73 |ram.section by s| 00000430 65 63 74 69 6f 6e 20 61 6e 64 20 62 72 69 65 66 |ection and brief| 00000440 6c 79 20 64 69 73 63 75 73 73 20 74 68 65 20 70 |ly discuss the p| 00000450 72 6f 62 6c 65 6d 73 20 49 20 63 61 6d 65 0d 61 |roblems I came.a| 00000460 63 72 6f 73 73 20 69 6e 20 77 72 69 74 69 6e 67 |cross in writing| 00000470 20 69 74 2e 0d 0d 41 73 20 77 69 74 68 20 74 68 | it...As with th| 00000480 65 20 6c 61 73 74 20 6d 6f 64 75 6c 65 2c 20 74 |e last module, t| 00000490 68 69 73 20 70 72 6f 67 72 61 6d 20 69 73 20 64 |his program is d| 000004a0 69 76 69 64 65 64 20 69 6e 74 6f 0d 73 75 62 72 |ivided into.subr| 000004b0 6f 75 74 69 6e 65 73 20 74 68 61 74 20 63 61 72 |outines that car| 000004c0 72 79 20 6f 75 74 20 74 68 65 20 6d 61 69 6e 20 |ry out the main | 000004d0 73 65 63 74 69 6f 6e 73 20 6f 66 20 74 68 65 20 |sections of the | 000004e0 77 6f 72 6b 2e 20 0d 4f 6e 65 20 73 69 67 6e 69 |work. .One signi| 000004f0 66 69 63 61 6e 74 20 64 69 66 66 65 72 65 6e 63 |ficant differenc| 00000500 65 20 69 73 20 74 68 61 74 2c 20 74 68 69 73 20 |e is that, this | 00000510 74 69 6d 65 2c 20 49 20 68 61 76 65 20 75 73 65 |time, I have use| 00000520 64 0d 43 41 4c 4c 20 77 69 74 68 20 70 61 72 61 |d.CALL with para| 00000530 6d 65 74 65 72 73 20 61 73 20 61 20 6d 65 61 6e |meters as a mean| 00000540 73 20 6f 66 20 67 65 74 74 69 6e 67 20 74 68 65 |s of getting the| 00000550 20 66 6c 6f 61 74 69 6e 67 0d 70 6f 69 6e 74 20 | floating.point | 00000560 6e 75 6d 62 65 72 73 20 69 6e 74 6f 20 74 68 65 |numbers into the| 00000570 20 70 72 6f 67 72 61 6d 2e 20 20 54 68 69 73 20 | program. This | 00000580 6d 65 61 6e 73 20 74 68 61 74 20 74 68 65 20 63 |means that the c| 00000590 6f 64 65 20 69 73 0d 6f 6e 6c 79 20 61 73 73 65 |ode is.only asse| 000005a0 6d 62 6c 65 64 20 6f 6e 63 65 20 69 6e 20 65 61 |mbled once in ea| 000005b0 63 68 20 52 55 4e 20 77 69 74 68 20 61 20 52 45 |ch RUN with a RE| 000005c0 50 45 41 54 20 55 4e 54 49 4c 20 46 41 4c 53 45 |PEAT UNTIL FALSE| 000005d0 0d 6c 6f 6f 70 20 6c 65 74 74 69 6e 67 20 79 6f |.loop letting yo| 000005e0 75 20 65 6e 74 65 72 20 6e 65 77 20 76 61 6c 75 |u enter new valu| 000005f0 65 73 20 61 6e 64 20 73 65 65 69 6e 67 20 74 68 |es and seeing th| 00000600 65 20 72 65 73 75 6c 74 73 0d 65 61 63 68 20 74 |e results.each t| 00000610 69 6d 65 2e 20 20 54 68 69 73 20 69 73 20 61 6e |ime. This is an| 00000620 20 61 6c 74 65 72 6e 61 74 69 76 65 20 74 6f 20 | alternative to | 00000630 75 73 69 6e 67 20 74 68 65 20 46 4e 45 51 55 46 |using the FNEQUF| 00000640 0d 70 73 65 75 64 6f 2d 6f 70 74 20 69 6e 20 74 |.pseudo-opt in t| 00000650 68 65 20 61 73 73 65 6d 62 6c 65 72 2e 20 20 59 |he assembler. Y| 00000660 6f 75 20 6d 61 79 20 6c 69 6b 65 20 74 6f 20 6d |ou may like to m| 00000670 6f 64 69 66 79 20 74 68 65 0d 6c 61 73 74 20 6d |odify the.last m| 00000680 6f 64 75 6c 65 20 74 6f 20 77 6f 72 6b 20 69 6e |odule to work in| 00000690 20 74 68 69 73 20 77 61 79 2e 0d 0d 48 61 76 69 | this way...Havi| 000006a0 6e 67 20 62 65 65 6e 20 74 72 61 6e 73 66 65 72 |ng been transfer| 000006b0 72 65 64 2c 20 75 73 69 6e 67 20 69 6e 66 6f 72 |red, using infor| 000006c0 6d 61 74 69 6f 6e 20 69 6e 20 74 68 65 20 70 61 |mation in the pa| 000006d0 72 61 6d 65 74 65 72 0d 62 6c 6f 63 6b 20 61 74 |rameter.block at| 000006e0 20 26 36 30 30 20 74 68 61 74 20 42 41 53 49 43 | &600 that BASIC| 000006f0 20 73 65 74 73 20 75 70 20 66 6f 72 20 43 41 4c | sets up for CAL| 00000700 4c 20 77 69 74 68 20 70 61 72 61 6d 65 74 65 72 |L with parameter| 00000710 73 2c 0d 74 68 65 20 6e 75 6d 62 65 72 73 20 69 |s,.the numbers i| 00000720 6e 70 75 74 20 61 72 65 20 73 74 6f 72 65 64 20 |nput are stored | 00000730 73 65 63 75 72 65 6c 79 2c 20 73 69 6e 63 65 20 |securely, since | 00000740 74 68 65 79 20 77 69 6c 6c 20 62 65 0d 6e 65 65 |they will be.nee| 00000750 64 65 64 20 74 77 69 63 65 2c 20 6f 6e 63 65 20 |ded twice, once | 00000760 66 6f 72 20 6d 75 6c 74 69 70 6c 69 63 61 74 69 |for multiplicati| 00000770 6f 6e 20 61 6e 64 20 6f 6e 63 65 20 66 6f 72 20 |on and once for | 00000780 64 69 76 69 73 69 6f 6e 2e 20 0d 4e 6f 74 65 20 |division. .Note | 00000790 74 68 61 74 20 74 68 65 20 69 6e 69 74 69 61 6c |that the initial| 000007a0 20 74 72 61 6e 73 66 65 72 20 6e 65 65 64 73 20 | transfer needs | 000007b0 73 6f 6d 65 20 7a 65 72 6f 20 70 61 67 65 0d 77 |some zero page.w| 000007c0 6f 72 6b 73 70 61 63 65 20 62 65 63 61 75 73 65 |orkspace because| 000007d0 20 6f 66 20 74 68 65 20 69 6e 64 69 72 65 63 74 | of the indirect| 000007e0 20 61 64 64 72 65 73 73 69 6e 67 20 75 73 65 64 | addressing used| 000007f0 2e 20 20 54 68 65 0d 62 6c 6f 63 6b 20 61 74 20 |. The.block at | 00000800 26 36 30 30 20 6c 6f 6f 6b 73 20 6c 69 6b 65 20 |&600 looks like | 00000810 74 68 69 73 3a 0d 0d 20 20 20 20 26 36 30 30 20 |this:.. &600 | 00000820 20 20 20 31 20 62 79 74 65 20 2d 20 6e 75 6d 62 | 1 byte - numb| 00000830 65 72 20 6f 66 20 70 61 72 61 6d 65 74 65 72 73 |er of parameters| 00000840 0d 20 20 20 20 26 36 30 31 20 20 20 20 32 20 62 |. &601 2 b| 00000850 79 74 65 73 20 2d 20 70 61 72 61 6d 65 74 65 72 |ytes - parameter| 00000860 20 61 64 64 72 65 73 73 0d 20 20 20 20 26 36 30 | address. &60| 00000870 33 20 20 20 20 31 20 62 79 74 65 20 2d 20 70 61 |3 1 byte - pa| 00000880 72 61 6d 65 74 65 72 20 74 79 70 65 0d 20 20 20 |rameter type. | 00000890 20 26 36 30 34 20 20 20 20 32 20 62 79 74 65 73 | &604 2 bytes| 000008a0 20 2d 20 6e 65 78 74 20 70 61 72 61 6d 65 74 65 | - next paramete| 000008b0 72 20 61 64 64 72 65 73 73 20 2e 2e 2e 2e 20 61 |r address .... a| 000008c0 6e 64 20 73 6f 20 6f 6e 0d 0d 54 68 65 20 70 61 |nd so on..The pa| 000008d0 72 61 6d 65 74 65 72 20 74 79 70 65 20 66 6f 72 |rameter type for| 000008e0 20 61 20 66 6c 6f 61 74 69 6e 67 20 70 6f 69 6e | a floating poin| 000008f0 74 20 76 61 72 69 61 62 6c 65 20 69 73 20 35 2e |t variable is 5.| 00000900 20 0d 4f 74 68 65 72 73 20 61 72 65 20 30 20 66 | .Others are 0 f| 00000910 6f 72 20 61 20 62 79 74 65 20 28 69 2e 65 2e 20 |or a byte (i.e. | 00000920 61 6e 20 3f 58 29 2c 20 34 20 66 6f 72 20 61 6e |an ?X), 4 for an| 00000930 20 69 6e 74 65 67 65 72 2c 20 31 32 38 0d 66 6f | integer, 128.fo| 00000940 72 20 61 20 24 58 20 73 74 72 69 6e 67 20 69 6e |r a $X string in| 00000950 20 6d 65 6d 6f 72 79 20 28 74 65 72 6d 69 6e 61 | memory (termina| 00000960 74 65 64 20 62 79 20 61 20 43 52 29 20 61 6e 64 |ted by a CR) and| 00000970 20 31 32 39 20 66 6f 72 20 61 0d 73 74 72 69 6e | 129 for a.strin| 00000980 67 20 76 61 72 69 61 62 6c 65 20 28 58 24 29 2e |g variable (X$).| 00000990 20 20 49 6e 20 74 68 65 20 6c 61 73 74 20 63 61 | In the last ca| 000009a0 73 65 20 28 73 65 65 20 6d 6f 64 75 6c 65 20 37 |se (see module 7| 000009b0 29 20 74 68 65 0d 70 61 72 61 6d 65 74 65 72 20 |) the.parameter | 000009c0 61 64 64 72 65 73 73 20 69 73 20 61 20 70 6f 69 |address is a poi| 000009d0 6e 74 65 72 20 74 6f 20 61 20 73 74 72 69 6e 67 |nter to a string| 000009e0 20 69 6e 66 6f 72 6d 61 74 69 6f 6e 0d 62 6c 6f | information.blo| 000009f0 63 6b 2e 0d 0d 54 68 65 20 6e 75 6d 62 65 72 73 |ck...The numbers| 00000a00 20 61 72 65 20 74 68 65 6e 20 74 72 61 6e 73 66 | are then transf| 00000a10 65 72 72 65 64 20 69 6e 74 6f 20 74 68 65 20 77 |erred into the w| 00000a20 6f 72 6b 73 70 61 63 65 73 20 66 6f 72 20 74 68 |orkspaces for th| 00000a30 65 0d 63 61 6c 63 75 6c 61 74 69 6f 6e 20 62 79 |e.calculation by| 00000a40 20 61 20 73 75 62 72 6f 75 74 69 6e 65 20 27 74 | a subroutine 't| 00000a50 72 61 6e 73 66 65 72 5f 69 6e 27 2e 20 20 54 68 |ransfer_in'. Th| 00000a60 65 0d 6d 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e |e.multiplication| 00000a70 20 69 73 20 63 61 72 72 69 65 64 20 6f 75 74 20 | is carried out | 00000a80 61 6e 64 20 74 68 65 6e 20 74 68 65 20 72 65 73 |and then the res| 00000a90 75 6c 74 20 69 73 0d 74 72 61 6e 73 66 65 72 72 |ult is.transferr| 00000aa0 65 64 20 6f 75 74 20 69 6e 74 6f 20 27 72 65 73 |ed out into 'res| 00000ab0 75 6c 74 5f 6d 75 6c 74 27 2e 20 20 54 68 69 73 |ult_mult'. This| 00000ac0 20 70 72 6f 63 65 73 73 20 69 73 0d 72 65 70 65 | process is.repe| 00000ad0 61 74 65 64 20 66 6f 72 20 74 68 65 20 64 69 76 |ated for the div| 00000ae0 69 73 69 6f 6e 2e 20 20 41 74 20 74 68 65 20 65 |ision. At the e| 00000af0 6e 64 20 74 68 65 20 72 65 73 75 6c 74 73 20 6f |nd the results o| 00000b00 66 20 74 68 65 0d 6d 61 63 68 69 6e 65 20 63 6f |f the.machine co| 00000b10 64 65 20 63 61 6c 63 75 6c 61 74 69 6f 6e 73 20 |de calculations | 00000b20 61 72 65 20 63 6f 6d 70 61 72 65 64 20 77 69 74 |are compared wit| 00000b30 68 20 74 68 65 20 72 65 73 75 6c 74 20 75 73 69 |h the result usi| 00000b40 6e 67 0d 42 42 43 20 42 41 53 49 43 2e 20 20 49 |ng.BBC BASIC. I| 00000b50 20 77 69 6c 6c 20 61 64 6d 69 74 20 68 65 72 65 | will admit here| 00000b60 20 61 6e 64 20 6e 6f 77 20 74 68 61 74 20 6d 79 | and now that my| 00000b70 20 72 6f 75 74 69 6e 65 73 20 61 72 65 20 61 0d | routines are a.| 00000b80 6c 69 74 74 6c 65 20 69 6e 6e 61 63 75 72 61 74 |little innacurat| 00000b90 65 20 62 75 74 20 49 20 77 69 6c 6c 20 61 6c 73 |e but I will als| 00000ba0 6f 20 61 64 6d 69 74 20 74 6f 20 74 68 69 73 20 |o admit to this | 00000bb0 62 65 69 6e 67 20 6f 66 20 61 6e 0d 65 78 70 65 |being of an.expe| 00000bc0 72 69 6d 65 6e 74 61 6c 20 6e 61 74 75 72 65 20 |rimental nature | 00000bd0 61 6e 64 20 49 20 77 61 73 20 74 72 79 69 6e 67 |and I was trying| 00000be0 20 66 6f 72 20 63 6c 61 72 69 74 79 2e 0d 0d 41 | for clarity...A| 00000bf0 73 20 77 65 20 67 6f 20 64 6f 77 6e 20 74 68 65 |s we go down the| 00000c00 20 6c 69 73 74 69 6e 67 20 6f 66 20 74 68 65 20 | listing of the | 00000c10 61 73 73 65 6d 62 6c 79 20 70 72 6f 67 72 61 6d |assembly program| 00000c20 20 74 68 65 20 6e 65 78 74 0d 74 68 69 6e 67 20 | the next.thing | 00000c30 77 65 20 6d 65 65 74 20 69 73 20 61 20 62 6c 6f |we meet is a blo| 00000c40 63 6b 20 6f 66 20 77 6f 72 6b 73 70 61 63 65 2e |ck of workspace.| 00000c50 20 20 49 20 68 61 76 65 20 69 6e 63 72 65 61 73 | I have increas| 00000c60 65 64 20 74 68 65 0d 77 6f 72 6b 73 70 61 63 65 |ed the.workspace| 00000c70 20 66 6f 72 20 74 68 65 20 63 61 6c 63 75 6c 61 | for the calcula| 00000c80 74 69 6f 6e 73 20 74 6f 20 73 65 76 65 6e 20 62 |tions to seven b| 00000c90 79 74 65 73 20 66 72 6f 6d 20 66 69 76 65 0d 62 |ytes from five.b| 00000ca0 65 63 61 75 73 65 20 49 20 6e 65 65 64 20 73 65 |ecause I need se| 00000cb0 76 65 6e 20 66 6f 72 20 64 69 76 69 73 69 6f 6e |ven for division| 00000cc0 2e 20 20 49 20 68 61 76 65 20 61 6c 73 6f 20 74 |. I have also t| 00000cd0 72 69 70 6c 65 20 6e 61 6d 65 64 0d 27 66 70 77 |riple named.'fpw| 00000ce0 73 5f 31 27 20 61 6e 64 20 27 66 70 77 73 5f 32 |s_1' and 'fpws_2| 00000cf0 27 20 66 6f 72 20 74 68 65 69 72 20 6d 75 6c 74 |' for their mult| 00000d00 69 70 6c 69 63 61 74 69 6f 6e 20 61 6e 64 20 64 |iplication and d| 00000d10 69 76 69 73 69 6f 6e 0d 72 6f 6c 65 73 2e 20 20 |ivision.roles. | 00000d20 54 68 69 73 20 69 73 20 61 20 74 65 63 68 6e 69 |This is a techni| 00000d30 71 75 65 20 79 6f 75 20 63 61 6e 20 75 73 65 20 |que you can use | 00000d40 69 66 20 79 6f 75 20 77 61 6e 74 20 74 6f 20 62 |if you want to b| 00000d50 65 0d 63 6c 65 61 72 20 61 62 6f 75 74 20 74 68 |e.clear about th| 00000d60 65 20 66 75 6e 63 74 69 6f 6e 20 6f 66 20 61 20 |e function of a | 00000d70 70 69 65 63 65 20 6f 66 20 6d 65 6d 6f 72 79 20 |piece of memory | 00000d80 61 6c 74 68 6f 75 67 68 20 69 74 73 0d 75 73 65 |although its.use| 00000d90 20 63 68 61 6e 67 65 73 20 64 75 72 69 6e 67 20 | changes during | 00000da0 74 68 65 20 70 72 6f 67 72 61 6d 2e 20 20 54 68 |the program. Th| 00000db0 65 72 65 20 69 73 20 6e 6f 20 70 65 6e 61 6c 74 |ere is no penalt| 00000dc0 79 20 66 6f 72 0d 6d 75 6c 74 69 2d 6e 61 6d 69 |y for.multi-nami| 00000dd0 6e 67 2c 20 61 70 61 72 74 20 66 72 6f 6d 20 6d |ng, apart from m| 00000de0 61 6b 69 6e 67 20 74 68 65 20 61 73 73 65 6d 62 |aking the assemb| 00000df0 6c 79 20 73 6f 75 72 63 65 20 63 6f 64 65 20 61 |ly source code a| 00000e00 0d 62 69 74 20 6c 6f 6e 67 65 72 2e 20 20 59 6f |.bit longer. Yo| 00000e10 75 72 20 66 69 6e 61 6c 20 70 72 6f 67 72 61 6d |ur final program| 00000e20 20 63 6f 64 65 2c 20 77 68 69 63 68 20 64 6f 65 | code, which doe| 00000e30 73 20 6e 6f 74 20 75 73 65 20 74 68 65 0d 6c 61 |s not use the.la| 00000e40 62 65 6c 73 2c 20 69 73 20 6e 6f 74 20 61 66 66 |bels, is not aff| 00000e50 65 63 74 65 64 2e 0d 0d 53 6f 20 6c 65 74 27 73 |ected...So let's| 00000e60 20 6c 6f 6f 6b 20 61 74 20 74 68 65 20 6d 75 6c | look at the mul| 00000e70 74 69 70 6c 69 63 61 74 69 6f 6e 20 66 69 72 73 |tiplication firs| 00000e80 74 2e 0d 0d 54 6f 20 61 76 6f 69 64 20 70 72 6f |t...To avoid pro| 00000e90 62 6c 65 6d 73 20 49 20 68 61 76 65 20 74 72 61 |blems I have tra| 00000ea0 70 70 65 64 20 7a 65 72 6f 73 20 61 74 20 74 68 |pped zeros at th| 00000eb0 65 20 73 74 61 72 74 20 6f 66 20 74 68 65 0d 6d |e start of the.m| 00000ec0 75 6c 74 69 70 6c 79 69 6e 67 20 72 6f 75 74 69 |ultiplying routi| 00000ed0 6e 65 73 2e 20 20 4f 62 76 69 6f 75 73 6c 79 20 |nes. 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OS\BITS/T\OSB23.m0
OS\BITS/T\OSB23.m1
OS\BITS/T\OSB23.m2
OS\BITS/T\OSB23.m4
OS\BITS/T\OSB23.m5
.