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OS\BITS/T\OSB22
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.
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File contents
OSBITS - An Exploration of the BBC Micro at Machine Level
By Programmer
..........................................................
Part 22: Floating Point Arithmetic
In the last module we used a fixed point arithmetic system
for calculations. The result was relatively fast but was
slightly innacurate since very small numbers were not
represented as well as they could have been.
You will remember that we converted from a real number to a
fixed point number by multiplying by a constant factor. The
decimal example was that 12.34 became 12340 if the constant
factor was 100, and that .0123 became 12 (the last digit
being lost). All our calculating was then done on the new,
larger numbers, which were integers and so we could use
integer arithmetic. When we had our integer result it could
be converted back to its proper size by dividing by the
constant factor. We also needed to know how multilplying
and dividing would be affected by this factor and compensate
for it.
The next logical step in this numbering is to make the
factor variable and to keep track of what it is. This
could, for example, make 12.34 be represented by any of the
following:
1.234 * 10^+1
12.34 * 10^+0
123.4 * 10^-1
1234 * 10^-2
the last digit being a power of 10. This is the scientific
notation used in the BBC Micro but written as E-2 etc. This
part of the number is called the EXPONENT. Since all the
above list are equivalent there is a need to decide on a
normal format for such notation. There is a standard that
says that the first number (the MANTISSA) nust always be
between 0.1 and 1. This would lead to 12.34 being written
as 0.1234E2 (meaning multiplied by 10 to the power 2). The
BBC Micro standard has the mantissa between 1 and 10 so that
12.34 would be 1.234E1 (meaning multiplied by 10 to the
power 1).
Here we have a floating point decimal numbering convention
such that the mantissa is always of a size between two
limits and the exponent is the power of 10 by which you
multiply the mantissa to produce the required number. You
will notice straight away that the mantissa is not an
integer. Does this matter? In fact it does not.
The object with our fixed point system in the last program
modules was to work with integers in the knowledge that our
numbers were in reality 65536 times too large. But they
were only 65536 times too large if we insisted in thinking
of them as integers in this way. The alternative way was to
put a ficticious 'decimal' point between the top two bytes
of the number and the bottom two. This method would have
worked equally well and is essentially the same, it was only
a question of how we visualised the numbers being stored.
The rules for arithmetic for floating point are similar to
those for fixed point, except that the constant factor is
now variable and is known as the exponent.
Addition requires that the exponent of the two numbers be
the same, then the mantissae are added. You keep the size
of the number the same by multiplying the mantissa by 10 to
the power of the number you subtract from the exponent (or
vice versa).
For example
1.23E5 + 2.34E3
setting both to have an exponent of E3
= 123E3 + 2.34E3
= 125.34E3
= 1.2534E5 (re-normalised)
It is up to you which number you de-normalise to match
which. On paper it is often clearer to avoid too many zeros
by de-normalising the larger number.
Subtraction is similar to addition in that the exponents
must be made the same before subtraction, then normalisation
is carried out.
Multiplication is done by multiplying the mantissae and
adding the exponents.
For example
1.23E5 * 2.34E3
= (1.23*2.34)E8
= 2.8782E8
In this case we don't need to renormalise.
Division is similar to multiplication in that we divide the
mantissae and subtract the exponents.
For example
1.23E5 / 2.34E3
= (1.23/2.34)E2
= 0.5256E2
= 5.256E1 approximately
Incidentally, the habit of working with mantissae between 0
and 1 or between 1 and 10 dates back to the days of slide
rules. The slide rule would take care of the calculations
of the mantissae but it was up to you to sort out the
position of your decimal point.
BBC BASIC has a convention for floating point arithmetic
which we will use in this module. Using the exisiting
convention means I can use BBC BASIC to translate from
decimal into floating point instead of writing a conversion
routine. This could be done by modifying the ASCII to
binary routine in module 12.
Five bytes are used to store a number, four bytes of
mantissa and one of exponent.
The binary equivalent of of 3.25 is 11.01 since, after the
binary point, the first place is 'halves' the second is
'quarters', the third is 'eighths' and so on.
To normalise this number under the BBC BASIC conventionwe
must have the binary point at the head of the number such
that
3.25 (decimal) = 11.01 (binary) = .1101 x2^2
This gives us a representation in memory like this
Exponent Mantissa
--------------------------------------------------------
| 00000010 | 11010000 | 00000000 | 00000000 | 00000000 |
--------------------------------------------------------
since the exponent is 2 (10 in binary) and the mantissa is
aligned to the left. In this case, like one of our
conventions for decimal floating point, the mantissa is
always between 0 and 1. In effect you shift the mantissa
until its top bit is set. This means you can check the
negative flag (which reflects bit 7) of the first mantissa
byte if you want to check for normalisation.
It is vital to remember that, unlike integer numbers, where
the most significant byte is stored at the greatest address,
with floating point in the BBC Micro the exponent is at the
lowest address and the mantissa is stored in the next four
higher bytes, with the most significant byte in the LOWEST
address.
Now, because the first bit of the mantissa is ALWAYS going
to be set, unless the number is zero, BASIC uses that top
bit to denote sign. This is not a 2's complement notation,
it is purely and simply a sign bit. If it is set the number
is positive, if it is clear the number is negative. But
remember that this is using a bit that ought to be set as a
flag and in fact the top bit is actually always set.
One final contrivance of BBC BASIC is to add &80 to the
exponent to aid in calculating, so I have done that below.
Hence, this is the representation of 3.25
Exponent Mantissa
--------------------------------------------------------
| 10000010 | 01010000 | 00000000 | 00000000 | 00000000 |
--------------------------------------------------------
&82 &50 &00 &00 &00
and this of -3.25
Exponent Mantissa
--------------------------------------------------------
| 10000010 | 11010000 | 00000000 | 00000000 | 00000000 |
--------------------------------------------------------
&82 &D0 &00 &00 &00
Zero is stored as five zero bytes.
In calculating with floating point numbers of this format we
have three operations. Firstly we carry out the calculation
on the mantissae, secondly we carry out the operation on the
exponents and finally we normalise the result.
The program in this module takes a pair of numbers you enter
and calculates the sum and difference of them using modified
versions of the multi-byte arithmetical routines I have
introduced earlier in the series.
I won't go into great detail here about the program since it
is copiously annotated but a few points are worth
mentioning.
The input of the floating point numbers is done using a
pseudo OPT function defined at the end of the program. It
relies on us knowing where BASIC will store the first
variable beginning with a ` sign (pound). We can use this
knowledge to both put and retrieve BBC BASIC format floating
point numbers.
Ironically, although having separately signed numbers makes
life easier for multiplication and division (next module) it
actually complicates addition and subtraction more than
somewhat. I have introduced an extra byte to act as an
overflow with each of the numbers in the workspace so that
we can convert negative numbers to 2's complement and back
again. In fact the BASIC fp routines also use a rounding
byte to improve precision but that does not seem to be
necessary here.
The main routines for addition and subtraction are
straightforward enough. They are basically five byte
multiple precision similar to those in earlier modules. The
curious difference is that, as I mentioned before, the bytes
are stored back to front with the most significant byte in
the lowest memory location.
In the main section of the program there are these
subroutines:
'move_sign_bit' moves the sign bits from the top of the
mantissae and puts them into a separate byte for later use.
'equate_exponents' denormalises the smaller of the two
numbers so that the exponents of the two numbers are the
same. It does this by rotating the mantissa right (which
makes it smaller) while increasing the exponent until the
exponents match.
'neg_convert' makes any negative numbers into 2's
complement, making use of the extra overflow byte.
Then come the addition or subtraction subroutines.
'neg_convert_back' takes any result that is 2's complement
negative and makes it positive while modifying its sign
byte.
Then we trap a zero result since the zero format is
non-standard and would put the renormalising routine into an
endless loop.
'renormalise' adjusts the result until it is a normalised
number (with the top bit of the mantissa set) as expected by
BASIC. The exponent is adjusted accordingly. If there is
anything in the overflow byte then the whole mantissa
(including the overflow byte) is rotated right until the
overflow byte is empty. The exponent is increased by one
for each rotation. If, during the course of this increment,
the exponent becomes zero then we have generated a number
too large for the system to handle, and an error is
generated. It has the number 20 but I have given it a
different message to the standard 'too big'.
If the overflow byte is empty then the rest of the mantissa
is rotated left until the top bit of the mantissa is set.
The exponent is decreased by one for each rotation. Of
course if the top bit is already set then the number is
correct.
Finally the routine at 'replace_top_bit' calculates the new
sign bit and replaces the top bit of the mantissa with it.
When you run the program you may have to be in MODE 7
because the assember code is rather long. You enter a
couple of numbers of virtually any size, and you will be
able to compare the addition and subtraction results from
this machine code and from BASIC. The machine code here is
almost twice as fast as BASIC although in some circumstances
it is not quite as precise. This is only likely to show as
a difference of 1 in the smallest significant digit between
the two proffered results.
Next time multiplication and division.
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 32 3a 20 |.......Part 22: | 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 0d 0d 0d 49 6e 20 74 |rithmetic...In t| 000000b0 68 65 20 6c 61 73 74 20 6d 6f 64 75 6c 65 20 77 |he last module w| 000000c0 65 20 75 73 65 64 20 61 20 66 69 78 65 64 20 70 |e used a fixed p| 000000d0 6f 69 6e 74 20 61 72 69 74 68 6d 65 74 69 63 20 |oint arithmetic | 000000e0 73 79 73 74 65 6d 0d 66 6f 72 20 63 61 6c 63 75 |system.for calcu| 000000f0 6c 61 74 69 6f 6e 73 2e 20 20 54 68 65 20 72 65 |lations. The re| 00000100 73 75 6c 74 20 77 61 73 20 72 65 6c 61 74 69 76 |sult was relativ| 00000110 65 6c 79 20 66 61 73 74 20 62 75 74 20 77 61 73 |ely fast but was| 00000120 0d 73 6c 69 67 68 74 6c 79 20 69 6e 6e 61 63 75 |.slightly innacu| 00000130 72 61 74 65 20 73 69 6e 63 65 20 76 65 72 79 20 |rate since very | 00000140 73 6d 61 6c 6c 20 6e 75 6d 62 65 72 73 20 77 65 |small numbers we| 00000150 72 65 20 6e 6f 74 0d 72 65 70 72 65 73 65 6e 74 |re not.represent| 00000160 65 64 20 61 73 20 77 65 6c 6c 20 61 73 20 74 68 |ed as well as th| 00000170 65 79 20 63 6f 75 6c 64 20 68 61 76 65 20 62 65 |ey could have be| 00000180 65 6e 2e 0d 0d 59 6f 75 20 77 69 6c 6c 20 72 65 |en...You will re| 00000190 6d 65 6d 62 65 72 20 74 68 61 74 20 77 65 20 63 |member that we c| 000001a0 6f 6e 76 65 72 74 65 64 20 66 72 6f 6d 20 61 20 |onverted from a | 000001b0 72 65 61 6c 20 6e 75 6d 62 65 72 20 74 6f 20 61 |real number to a| 000001c0 0d 66 69 78 65 64 20 70 6f 69 6e 74 20 6e 75 6d |.fixed point num| 000001d0 62 65 72 20 62 79 20 6d 75 6c 74 69 70 6c 79 69 |ber by multiplyi| 000001e0 6e 67 20 62 79 20 61 20 63 6f 6e 73 74 61 6e 74 |ng by a constant| 000001f0 20 66 61 63 74 6f 72 2e 20 20 54 68 65 0d 64 65 | factor. The.de| 00000200 63 69 6d 61 6c 20 65 78 61 6d 70 6c 65 20 77 61 |cimal example wa| 00000210 73 20 74 68 61 74 20 31 32 2e 33 34 20 62 65 63 |s that 12.34 bec| 00000220 61 6d 65 20 31 32 33 34 30 20 69 66 20 74 68 65 |ame 12340 if the| 00000230 20 63 6f 6e 73 74 61 6e 74 0d 66 61 63 74 6f 72 | constant.factor| 00000240 20 77 61 73 20 31 30 30 2c 20 61 6e 64 20 74 68 | was 100, and th| 00000250 61 74 20 2e 30 31 32 33 20 62 65 63 61 6d 65 20 |at .0123 became | 00000260 31 32 20 28 74 68 65 20 6c 61 73 74 20 64 69 67 |12 (the last dig| 00000270 69 74 0d 62 65 69 6e 67 20 6c 6f 73 74 29 2e 20 |it.being lost). | 00000280 20 41 6c 6c 20 6f 75 72 20 63 61 6c 63 75 6c 61 | All our calcula| 00000290 74 69 6e 67 20 77 61 73 20 74 68 65 6e 20 64 6f |ting was then do| 000002a0 6e 65 20 6f 6e 20 74 68 65 20 6e 65 77 2c 0d 6c |ne on the new,.l| 000002b0 61 72 67 65 72 20 6e 75 6d 62 65 72 73 2c 20 77 |arger numbers, w| 000002c0 68 69 63 68 20 77 65 72 65 20 69 6e 74 65 67 65 |hich were intege| 000002d0 72 73 20 61 6e 64 20 73 6f 20 77 65 20 63 6f 75 |rs and so we cou| 000002e0 6c 64 20 75 73 65 0d 69 6e 74 65 67 65 72 20 61 |ld use.integer a| 000002f0 72 69 74 68 6d 65 74 69 63 2e 20 20 57 68 65 6e |rithmetic. When| 00000300 20 77 65 20 68 61 64 20 6f 75 72 20 69 6e 74 65 | we had our inte| 00000310 67 65 72 20 72 65 73 75 6c 74 20 69 74 20 63 6f |ger result it co| 00000320 75 6c 64 0d 62 65 20 63 6f 6e 76 65 72 74 65 64 |uld.be converted| 00000330 20 62 61 63 6b 20 74 6f 20 69 74 73 20 70 72 6f | back to its pro| 00000340 70 65 72 20 73 69 7a 65 20 62 79 20 64 69 76 69 |per size by divi| 00000350 64 69 6e 67 20 62 79 20 74 68 65 0d 63 6f 6e 73 |ding by the.cons| 00000360 74 61 6e 74 20 66 61 63 74 6f 72 2e 20 20 57 65 |tant factor. We| 00000370 20 61 6c 73 6f 20 6e 65 65 64 65 64 20 74 6f 20 | also needed to | 00000380 6b 6e 6f 77 20 68 6f 77 20 6d 75 6c 74 69 6c 70 |know how multilp| 00000390 6c 79 69 6e 67 0d 61 6e 64 20 64 69 76 69 64 69 |lying.and dividi| 000003a0 6e 67 20 77 6f 75 6c 64 20 62 65 20 61 66 66 65 |ng would be affe| 000003b0 63 74 65 64 20 62 79 20 74 68 69 73 20 66 61 63 |cted by this fac| 000003c0 74 6f 72 20 61 6e 64 20 63 6f 6d 70 65 6e 73 61 |tor and compensa| 000003d0 74 65 0d 66 6f 72 20 69 74 2e 0d 0d 54 68 65 20 |te.for it...The | 000003e0 6e 65 78 74 20 6c 6f 67 69 63 61 6c 20 73 74 65 |next logical ste| 000003f0 70 20 69 6e 20 74 68 69 73 20 6e 75 6d 62 65 72 |p in this number| 00000400 69 6e 67 20 69 73 20 74 6f 20 6d 61 6b 65 20 74 |ing is to make t| 00000410 68 65 0d 66 61 63 74 6f 72 20 76 61 72 69 61 62 |he.factor variab| 00000420 6c 65 20 61 6e 64 20 74 6f 20 6b 65 65 70 20 74 |le and to keep t| 00000430 72 61 63 6b 20 6f 66 20 77 68 61 74 20 69 74 20 |rack of what it | 00000440 69 73 2e 20 20 54 68 69 73 0d 63 6f 75 6c 64 2c |is. This.could,| 00000450 20 66 6f 72 20 65 78 61 6d 70 6c 65 2c 20 6d 61 | for example, ma| 00000460 6b 65 20 31 32 2e 33 34 20 62 65 20 72 65 70 72 |ke 12.34 be repr| 00000470 65 73 65 6e 74 65 64 20 62 79 20 61 6e 79 20 6f |esented by any o| 00000480 66 20 74 68 65 0d 66 6f 6c 6c 6f 77 69 6e 67 3a |f the.following:| 00000490 0d 0d 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |.. | 000004a0 20 20 20 20 20 20 31 2e 32 33 34 20 2a 20 31 30 | 1.234 * 10| 000004b0 5e 2b 31 0d 20 20 20 20 20 20 20 20 20 20 20 20 |^+1. | 000004c0 20 20 20 20 20 20 20 20 31 32 2e 33 34 20 2a 20 | 12.34 * | 000004d0 31 30 5e 2b 30 0d 20 20 20 20 20 20 20 20 20 20 |10^+0. | 000004e0 20 20 20 20 20 20 20 20 20 20 31 32 33 2e 34 20 | 123.4 | 000004f0 2a 20 31 30 5e 2d 31 0d 20 20 20 20 20 20 20 20 |* 10^-1. | 00000500 20 20 20 20 20 20 20 20 20 20 20 20 31 32 33 34 | 1234| 00000510 20 20 2a 20 31 30 5e 2d 32 0d 0d 74 68 65 20 6c | * 10^-2..the l| 00000520 61 73 74 20 64 69 67 69 74 20 62 65 69 6e 67 20 |ast digit being | 00000530 61 20 70 6f 77 65 72 20 6f 66 20 31 30 2e 20 20 |a power of 10. | 00000540 54 68 69 73 20 69 73 20 74 68 65 20 73 63 69 65 |This is the scie| 00000550 6e 74 69 66 69 63 0d 6e 6f 74 61 74 69 6f 6e 20 |ntific.notation | 00000560 75 73 65 64 20 69 6e 20 74 68 65 20 42 42 43 20 |used in the BBC | 00000570 4d 69 63 72 6f 20 62 75 74 20 77 72 69 74 74 65 |Micro but writte| 00000580 6e 20 61 73 20 45 2d 32 20 65 74 63 2e 20 20 54 |n as E-2 etc. T| 00000590 68 69 73 0d 70 61 72 74 20 6f 66 20 74 68 65 20 |his.part of the | 000005a0 6e 75 6d 62 65 72 20 69 73 20 63 61 6c 6c 65 64 |number is called| 000005b0 20 74 68 65 20 45 58 50 4f 4e 45 4e 54 2e 20 20 | the EXPONENT. | 000005c0 53 69 6e 63 65 20 61 6c 6c 20 74 68 65 0d 61 62 |Since all the.ab| 000005d0 6f 76 65 20 6c 69 73 74 20 61 72 65 20 65 71 75 |ove list are equ| 000005e0 69 76 61 6c 65 6e 74 20 74 68 65 72 65 20 69 73 |ivalent there is| 000005f0 20 61 20 6e 65 65 64 20 74 6f 20 64 65 63 69 64 | a need to decid| 00000600 65 20 6f 6e 20 61 0d 6e 6f 72 6d 61 6c 20 66 6f |e on a.normal fo| 00000610 72 6d 61 74 20 66 6f 72 20 73 75 63 68 20 6e 6f |rmat for such no| 00000620 74 61 74 69 6f 6e 2e 20 20 54 68 65 72 65 20 69 |tation. There i| 00000630 73 20 61 20 73 74 61 6e 64 61 72 64 20 74 68 61 |s a standard tha| 00000640 74 0d 73 61 79 73 20 74 68 61 74 20 74 68 65 20 |t.says that the | 00000650 66 69 72 73 74 20 6e 75 6d 62 65 72 20 28 74 68 |first number (th| 00000660 65 20 4d 41 4e 54 49 53 53 41 29 20 6e 75 73 74 |e MANTISSA) nust| 00000670 20 61 6c 77 61 79 73 20 62 65 0d 62 65 74 77 65 | always be.betwe| 00000680 65 6e 20 30 2e 31 20 61 6e 64 20 31 2e 20 20 54 |en 0.1 and 1. T| 00000690 68 69 73 20 77 6f 75 6c 64 20 6c 65 61 64 20 74 |his would lead t| 000006a0 6f 20 31 32 2e 33 34 20 62 65 69 6e 67 20 77 72 |o 12.34 being wr| 000006b0 69 74 74 65 6e 0d 61 73 20 30 2e 31 32 33 34 45 |itten.as 0.1234E| 000006c0 32 20 28 6d 65 61 6e 69 6e 67 20 6d 75 6c 74 69 |2 (meaning multi| 000006d0 70 6c 69 65 64 20 62 79 20 31 30 20 74 6f 20 74 |plied by 10 to t| 000006e0 68 65 20 70 6f 77 65 72 20 32 29 2e 20 20 54 68 |he power 2). Th| 000006f0 65 0d 42 42 43 20 4d 69 63 72 6f 20 73 74 61 6e |e.BBC Micro stan| 00000700 64 61 72 64 20 68 61 73 20 74 68 65 20 6d 61 6e |dard has the man| 00000710 74 69 73 73 61 20 62 65 74 77 65 65 6e 20 31 20 |tissa between 1 | 00000720 61 6e 64 20 31 30 20 73 6f 20 74 68 61 74 0d 31 |and 10 so that.1| 00000730 32 2e 33 34 20 77 6f 75 6c 64 20 62 65 20 31 2e |2.34 would be 1.| 00000740 32 33 34 45 31 20 28 6d 65 61 6e 69 6e 67 20 6d |234E1 (meaning m| 00000750 75 6c 74 69 70 6c 69 65 64 20 62 79 20 31 30 20 |ultiplied by 10 | 00000760 74 6f 20 74 68 65 0d 70 6f 77 65 72 20 31 29 2e |to the.power 1).| 00000770 0d 0d 48 65 72 65 20 77 65 20 68 61 76 65 20 61 |..Here we have a| 00000780 20 66 6c 6f 61 74 69 6e 67 20 70 6f 69 6e 74 20 | floating point | 00000790 64 65 63 69 6d 61 6c 20 6e 75 6d 62 65 72 69 6e |decimal numberin| 000007a0 67 20 63 6f 6e 76 65 6e 74 69 6f 6e 0d 73 75 63 |g convention.suc| 000007b0 68 20 74 68 61 74 20 74 68 65 20 6d 61 6e 74 69 |h that the manti| 000007c0 73 73 61 20 69 73 20 61 6c 77 61 79 73 20 6f 66 |ssa is always of| 000007d0 20 61 20 73 69 7a 65 20 62 65 74 77 65 65 6e 20 | a size between | 000007e0 74 77 6f 0d 6c 69 6d 69 74 73 20 61 6e 64 20 74 |two.limits and t| 000007f0 68 65 20 65 78 70 6f 6e 65 6e 74 20 69 73 20 74 |he exponent is t| 00000800 68 65 20 70 6f 77 65 72 20 6f 66 20 31 30 20 62 |he power of 10 b| 00000810 79 20 77 68 69 63 68 20 79 6f 75 0d 6d 75 6c 74 |y which you.mult| 00000820 69 70 6c 79 20 74 68 65 20 6d 61 6e 74 69 73 73 |iply the mantiss| 00000830 61 20 74 6f 20 70 72 6f 64 75 63 65 20 74 68 65 |a to produce the| 00000840 20 72 65 71 75 69 72 65 64 20 6e 75 6d 62 65 72 | required number| 00000850 2e 20 20 59 6f 75 0d 77 69 6c 6c 20 6e 6f 74 69 |. You.will noti| 00000860 63 65 20 73 74 72 61 69 67 68 74 20 61 77 61 79 |ce straight away| 00000870 20 74 68 61 74 20 74 68 65 20 6d 61 6e 74 69 73 | that the mantis| 00000880 73 61 20 69 73 20 6e 6f 74 20 61 6e 0d 69 6e 74 |sa is not an.int| 00000890 65 67 65 72 2e 20 20 44 6f 65 73 20 74 68 69 73 |eger. Does this| 000008a0 20 6d 61 74 74 65 72 3f 20 20 49 6e 20 66 61 63 | matter? In fac| 000008b0 74 20 69 74 20 64 6f 65 73 20 6e 6f 74 2e 0d 0d |t it does not...| 000008c0 54 68 65 20 6f 62 6a 65 63 74 20 77 69 74 68 20 |The object with | 000008d0 6f 75 72 20 66 69 78 65 64 20 70 6f 69 6e 74 20 |our fixed point | 000008e0 73 79 73 74 65 6d 20 69 6e 20 74 68 65 20 6c 61 |system in the la| 000008f0 73 74 20 70 72 6f 67 72 61 6d 0d 6d 6f 64 75 6c |st program.modul| 00000900 65 73 20 77 61 73 20 74 6f 20 77 6f 72 6b 20 77 |es was to work w| 00000910 69 74 68 20 69 6e 74 65 67 65 72 73 20 69 6e 20 |ith integers in | 00000920 74 68 65 20 6b 6e 6f 77 6c 65 64 67 65 20 74 68 |the knowledge th| 00000930 61 74 20 6f 75 72 0d 6e 75 6d 62 65 72 73 20 77 |at our.numbers w| 00000940 65 72 65 20 69 6e 20 72 65 61 6c 69 74 79 20 36 |ere in reality 6| 00000950 35 35 33 36 20 74 69 6d 65 73 20 74 6f 6f 20 6c |5536 times too l| 00000960 61 72 67 65 2e 20 20 42 75 74 20 74 68 65 79 0d |arge. But they.| 00000970 77 65 72 65 20 6f 6e 6c 79 20 36 35 35 33 36 20 |were only 65536 | 00000980 74 69 6d 65 73 20 74 6f 6f 20 6c 61 72 67 65 20 |times too large | 00000990 69 66 20 77 65 20 69 6e 73 69 73 74 65 64 20 69 |if we insisted i| 000009a0 6e 20 74 68 69 6e 6b 69 6e 67 0d 6f 66 20 74 68 |n thinking.of th| 000009b0 65 6d 20 61 73 20 69 6e 74 65 67 65 72 73 20 69 |em as integers i| 000009c0 6e 20 74 68 69 73 20 77 61 79 2e 20 20 54 68 65 |n this way. The| 000009d0 20 61 6c 74 65 72 6e 61 74 69 76 65 20 77 61 79 | alternative way| 000009e0 20 77 61 73 20 74 6f 0d 70 75 74 20 61 20 66 69 | was to.put a fi| 000009f0 63 74 69 63 69 6f 75 73 20 27 64 65 63 69 6d 61 |cticious 'decima| 00000a00 6c 27 20 70 6f 69 6e 74 20 62 65 74 77 65 65 6e |l' point between| 00000a10 20 74 68 65 20 74 6f 70 20 74 77 6f 20 62 79 74 | the top two byt| 00000a20 65 73 0d 6f 66 20 74 68 65 20 6e 75 6d 62 65 72 |es.of the number| 00000a30 20 61 6e 64 20 74 68 65 20 62 6f 74 74 6f 6d 20 | and the bottom | 00000a40 74 77 6f 2e 20 20 54 68 69 73 20 6d 65 74 68 6f |two. This metho| 00000a50 64 20 77 6f 75 6c 64 20 68 61 76 65 0d 77 6f 72 |d would have.wor| 00000a60 6b 65 64 20 65 71 75 61 6c 6c 79 20 77 65 6c 6c |ked equally well| 00000a70 20 61 6e 64 20 69 73 20 65 73 73 65 6e 74 69 61 | and is essentia| 00000a80 6c 6c 79 20 74 68 65 20 73 61 6d 65 2c 20 69 74 |lly the same, it| 00000a90 20 77 61 73 20 6f 6e 6c 79 0d 61 20 71 75 65 73 | was only.a ques| 00000aa0 74 69 6f 6e 20 6f 66 20 68 6f 77 20 77 65 20 76 |tion of how we v| 00000ab0 69 73 75 61 6c 69 73 65 64 20 74 68 65 20 6e 75 |isualised the nu| 00000ac0 6d 62 65 72 73 20 62 65 69 6e 67 20 73 74 6f 72 |mbers being stor| 00000ad0 65 64 2e 0d 0d 54 68 65 20 72 75 6c 65 73 20 66 |ed...The rules f| 00000ae0 6f 72 20 61 72 69 74 68 6d 65 74 69 63 20 66 6f |or arithmetic fo| 00000af0 72 20 66 6c 6f 61 74 69 6e 67 20 70 6f 69 6e 74 |r floating point| 00000b00 20 61 72 65 20 73 69 6d 69 6c 61 72 20 74 6f 0d | are similar to.| 00000b10 74 68 6f 73 65 20 66 6f 72 20 66 69 78 65 64 20 |those for fixed | 00000b20 70 6f 69 6e 74 2c 20 65 78 63 65 70 74 20 74 68 |point, except th| 00000b30 61 74 20 74 68 65 20 63 6f 6e 73 74 61 6e 74 20 |at the constant | 00000b40 66 61 63 74 6f 72 20 69 73 0d 6e 6f 77 20 76 61 |factor is.now va| 00000b50 72 69 61 62 6c 65 20 61 6e 64 20 69 73 20 6b 6e |riable and is kn| 00000b60 6f 77 6e 20 61 73 20 74 68 65 20 65 78 70 6f 6e |own as the expon| 00000b70 65 6e 74 2e 0d 0d 41 64 64 69 74 69 6f 6e 20 72 |ent...Addition r| 00000b80 65 71 75 69 72 65 73 20 74 68 61 74 20 74 68 65 |equires that the| 00000b90 20 65 78 70 6f 6e 65 6e 74 20 6f 66 20 74 68 65 | exponent of the| 00000ba0 20 74 77 6f 20 6e 75 6d 62 65 72 73 20 62 65 0d | two numbers be.| 00000bb0 74 68 65 20 73 61 6d 65 2c 20 74 68 65 6e 20 74 |the same, then t| 00000bc0 68 65 20 6d 61 6e 74 69 73 73 61 65 20 61 72 65 |he mantissae are| 00000bd0 20 61 64 64 65 64 2e 20 20 59 6f 75 20 6b 65 65 | added. You kee| 00000be0 70 20 74 68 65 20 73 69 7a 65 0d 6f 66 20 74 68 |p the size.of th| 00000bf0 65 20 6e 75 6d 62 65 72 20 74 68 65 20 73 61 6d |e number the sam| 00000c00 65 20 62 79 20 6d 75 6c 74 69 70 6c 79 69 6e 67 |e by multiplying| 00000c10 20 74 68 65 20 6d 61 6e 74 69 73 73 61 20 62 79 | the mantissa by| 00000c20 20 31 30 20 74 6f 0d 74 68 65 20 70 6f 77 65 72 | 10 to.the power| 00000c30 20 6f 66 20 74 68 65 20 6e 75 6d 62 65 72 20 79 | of the number y| 00000c40 6f 75 20 73 75 62 74 72 61 63 74 20 66 72 6f 6d |ou subtract from| 00000c50 20 74 68 65 20 65 78 70 6f 6e 65 6e 74 20 28 6f | the exponent (o| 00000c60 72 0d 76 69 63 65 20 76 65 72 73 61 29 2e 0d 0d |r.vice versa)...| 00000c70 46 6f 72 20 65 78 61 6d 70 6c 65 0d 20 20 20 20 |For example. | 00000c80 20 20 20 20 20 20 20 20 20 20 20 20 31 2e 32 33 | 1.23| 00000c90 45 35 20 2b 20 32 2e 33 34 45 33 0d 0d 73 65 74 |E5 + 2.34E3..set| 00000ca0 74 69 6e 67 20 62 6f 74 68 20 74 6f 20 68 61 76 |ting both to hav| 00000cb0 65 20 61 6e 20 65 78 70 6f 6e 65 6e 74 20 6f 66 |e an exponent of| 00000cc0 20 45 33 0d 0d 20 20 20 20 20 20 20 20 20 20 20 | E3.. | 00000cd0 20 20 3d 20 20 31 32 33 45 33 20 2b 20 32 2e 33 | = 123E3 + 2.3| 00000ce0 34 45 33 0d 20 20 20 20 20 20 20 20 20 20 20 20 |4E3. | 00000cf0 20 3d 20 20 31 32 35 2e 33 34 45 33 0d 20 20 20 | = 125.34E3. | 00000d00 20 20 20 20 20 20 20 20 20 20 3d 20 20 31 2e 32 | = 1.2| 00000d10 35 33 34 45 35 20 20 28 72 65 2d 6e 6f 72 6d 61 |534E5 (re-norma| 00000d20 6c 69 73 65 64 29 0d 0d 49 74 20 69 73 20 75 70 |lised)..It is up| 00000d30 20 74 6f 20 79 6f 75 20 77 68 69 63 68 20 6e 75 | to you which nu| 00000d40 6d 62 65 72 20 79 6f 75 20 64 65 2d 6e 6f 72 6d |mber you de-norm| 00000d50 61 6c 69 73 65 20 74 6f 20 6d 61 74 63 68 0d 77 |alise to match.w| 00000d60 68 69 63 68 2e 20 20 4f 6e 20 70 61 70 65 72 20 |hich. On paper | 00000d70 69 74 20 69 73 20 6f 66 74 65 6e 20 63 6c 65 61 |it is often clea| 00000d80 72 65 72 20 74 6f 20 61 76 6f 69 64 20 74 6f 6f |rer to avoid too| 00000d90 20 6d 61 6e 79 20 7a 65 72 6f 73 0d 62 79 20 64 | many zeros.by d| 00000da0 65 2d 6e 6f 72 6d 61 6c 69 73 69 6e 67 20 74 68 |e-normalising th| 00000db0 65 20 6c 61 72 67 65 72 20 6e 75 6d 62 65 72 2e |e larger number.| 00000dc0 0d 0d 53 75 62 74 72 61 63 74 69 6f 6e 20 69 73 |..Subtraction is| 00000dd0 20 73 69 6d 69 6c 61 72 20 74 6f 20 61 64 64 69 | similar to addi| 00000de0 74 69 6f 6e 20 69 6e 20 74 68 61 74 20 74 68 65 |tion in that the| 00000df0 20 65 78 70 6f 6e 65 6e 74 73 0d 6d 75 73 74 20 | exponents.must | 00000e00 62 65 20 6d 61 64 65 20 74 68 65 20 73 61 6d 65 |be made the same| 00000e10 20 62 65 66 6f 72 65 20 73 75 62 74 72 61 63 74 | before subtract| 00000e20 69 6f 6e 2c 20 74 68 65 6e 20 6e 6f 72 6d 61 6c |ion, then normal| 00000e30 69 73 61 74 69 6f 6e 0d 69 73 20 63 61 72 72 69 |isation.is carri| 00000e40 65 64 20 6f 75 74 2e 0d 0d 4d 75 6c 74 69 70 6c |ed out...Multipl| 00000e50 69 63 61 74 69 6f 6e 20 69 73 20 64 6f 6e 65 20 |ication is done | 00000e60 62 79 20 6d 75 6c 74 69 70 6c 79 69 6e 67 20 74 |by multiplying t| 00000e70 68 65 20 6d 61 6e 74 69 73 73 61 65 20 61 6e 64 |he mantissae and| 00000e80 0d 61 64 64 69 6e 67 20 74 68 65 20 65 78 70 6f |.adding the expo| 00000e90 6e 65 6e 74 73 2e 0d 0d 46 6f 72 20 65 78 61 6d |nents...For exam| 00000ea0 70 6c 65 0d 20 20 20 20 20 20 20 20 20 20 20 20 |ple. | 00000eb0 20 20 20 20 31 2e 32 33 45 35 20 2a 20 32 2e 33 | 1.23E5 * 2.3| 00000ec0 34 45 33 0d 20 20 20 20 20 20 20 20 20 20 20 20 |4E3. | 00000ed0 20 3d 20 20 28 31 2e 32 33 2a 32 2e 33 34 29 45 | = (1.23*2.34)E| 00000ee0 38 0d 20 20 20 20 20 20 20 20 20 20 20 20 20 3d |8. =| 00000ef0 20 20 32 2e 38 37 38 32 45 38 0d 0d 49 6e 20 74 | 2.8782E8..In t| 00000f00 68 69 73 20 63 61 73 65 20 77 65 20 64 6f 6e 27 |his case we don'| 00000f10 74 20 6e 65 65 64 20 74 6f 20 72 65 6e 6f 72 6d |t need to renorm| 00000f20 61 6c 69 73 65 2e 0d 0d 44 69 76 69 73 69 6f 6e |alise...Division| 00000f30 20 69 73 20 73 69 6d 69 6c 61 72 20 74 6f 20 6d | is similar to m| 00000f40 75 6c 74 69 70 6c 69 63 61 74 69 6f 6e 20 69 6e |ultiplication in| 00000f50 20 74 68 61 74 20 77 65 20 64 69 76 69 64 65 20 | that we divide | 00000f60 74 68 65 0d 6d 61 6e 74 69 73 73 61 65 20 61 6e |the.mantissae an| 00000f70 64 20 73 75 62 74 72 61 63 74 20 74 68 65 20 65 |d subtract the e| 00000f80 78 70 6f 6e 65 6e 74 73 2e 0d 0d 46 6f 72 20 65 |xponents...For e| 00000f90 78 61 6d 70 6c 65 0d 20 20 20 20 20 20 20 20 20 |xample. | 00000fa0 20 20 20 20 20 20 20 31 2e 32 33 45 35 20 2f 20 | 1.23E5 / | 00000fb0 32 2e 33 34 45 33 20 0d 20 20 20 20 20 20 20 20 |2.34E3 . | 00000fc0 20 20 20 20 20 3d 20 20 28 31 2e 32 33 2f 32 2e | = (1.23/2.| 00000fd0 33 34 29 45 32 0d 20 20 20 20 20 20 20 20 20 20 |34)E2. | 00000fe0 20 20 20 3d 20 20 30 2e 35 32 35 36 45 32 0d 20 | = 0.5256E2. | 00000ff0 20 20 20 20 20 20 20 20 20 20 20 20 3d 20 20 35 | = 5| 00001000 2e 32 35 36 45 31 20 20 20 61 70 70 72 6f 78 69 |.256E1 approxi| 00001010 6d 61 74 65 6c 79 0d 0d 49 6e 63 69 64 65 6e 74 |mately..Incident| 00001020 61 6c 6c 79 2c 20 74 68 65 20 68 61 62 69 74 20 |ally, the habit | 00001030 6f 66 20 77 6f 72 6b 69 6e 67 20 77 69 74 68 20 |of working with | 00001040 6d 61 6e 74 69 73 73 61 65 20 62 65 74 77 65 65 |mantissae betwee| 00001050 6e 20 30 0d 61 6e 64 20 31 20 6f 72 20 62 65 74 |n 0.and 1 or bet| 00001060 77 65 65 6e 20 31 20 61 6e 64 20 31 30 20 64 61 |ween 1 and 10 da| 00001070 74 65 73 20 62 61 63 6b 20 74 6f 20 74 68 65 20 |tes back to the | 00001080 64 61 79 73 20 6f 66 20 73 6c 69 64 65 0d 72 75 |days of slide.ru| 00001090 6c 65 73 2e 20 20 54 68 65 20 73 6c 69 64 65 20 |les. 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Usin| 00001180 67 20 74 68 65 20 65 78 69 73 69 74 69 6e 67 0d |g the exisiting.| 00001190 63 6f 6e 76 65 6e 74 69 6f 6e 20 6d 65 61 6e 73 |convention means| 000011a0 20 49 20 63 61 6e 20 75 73 65 20 42 42 43 20 42 | I can use BBC B| 000011b0 41 53 49 43 20 74 6f 20 74 72 61 6e 73 6c 61 74 |ASIC to translat| 000011c0 65 20 66 72 6f 6d 0d 64 65 63 69 6d 61 6c 20 69 |e from.decimal i| 000011d0 6e 74 6f 20 66 6c 6f 61 74 69 6e 67 20 70 6f 69 |nto floating poi| 000011e0 6e 74 20 69 6e 73 74 65 61 64 20 6f 66 20 77 72 |nt instead of wr| 000011f0 69 74 69 6e 67 20 61 20 63 6f 6e 76 65 72 73 69 |iting a conversi| 00001200 6f 6e 0d 72 6f 75 74 69 6e 65 2e 20 20 54 68 69 |on.routine. Thi| 00001210 73 20 63 6f 75 6c 64 20 62 65 20 64 6f 6e 65 20 |s could be done | 00001220 62 79 20 6d 6f 64 69 66 79 69 6e 67 20 74 68 65 |by modifying the| 00001230 20 41 53 43 49 49 20 74 6f 0d 62 69 6e 61 72 79 | ASCII to.binary| 00001240 20 72 6f 75 74 69 6e 65 20 69 6e 20 6d 6f 64 75 | routine in modu| 00001250 6c 65 20 31 32 2e 0d 0d 46 69 76 65 20 62 79 74 |le 12...Five byt| 00001260 65 73 20 61 72 65 20 75 73 65 64 20 74 6f 20 73 |es are used to s| 00001270 74 6f 72 65 20 61 20 6e 75 6d 62 65 72 2c 20 66 |tore a number, f| 00001280 6f 75 72 20 62 79 74 65 73 20 6f 66 0d 6d 61 6e |our bytes of.man| 00001290 74 69 73 73 61 20 61 6e 64 20 6f 6e 65 20 6f 66 |tissa and one of| 000012a0 20 65 78 70 6f 6e 65 6e 74 2e 0d 0d 54 68 65 20 | exponent...The | 000012b0 62 69 6e 61 72 79 20 65 71 75 69 76 61 6c 65 6e |binary equivalen| 000012c0 74 20 6f 66 20 6f 66 20 33 2e 32 35 20 69 73 20 |t of of 3.25 is | 000012d0 31 31 2e 30 31 20 73 69 6e 63 65 2c 20 61 66 74 |11.01 since, aft| 000012e0 65 72 20 74 68 65 0d 62 69 6e 61 72 79 20 70 6f |er the.binary po| 000012f0 69 6e 74 2c 20 74 68 65 20 66 69 72 73 74 20 70 |int, the first p| 00001300 6c 61 63 65 20 69 73 20 27 68 61 6c 76 65 73 27 |lace is 'halves'| 00001310 20 74 68 65 20 73 65 63 6f 6e 64 20 69 73 0d 27 | the second is.'| 00001320 71 75 61 72 74 65 72 73 27 2c 20 74 68 65 20 74 |quarters', the t| 00001330 68 69 72 64 20 69 73 20 27 65 69 67 68 74 68 73 |hird is 'eighths| 00001340 27 20 61 6e 64 20 73 6f 20 6f 6e 2e 0d 0d 54 6f |' and so on...To| 00001350 20 6e 6f 72 6d 61 6c 69 73 65 20 74 68 69 73 20 | normalise this | 00001360 6e 75 6d 62 65 72 20 75 6e 64 65 72 20 74 68 65 |number under the| 00001370 20 42 42 43 20 42 41 53 49 43 20 63 6f 6e 76 65 | BBC BASIC conve| 00001380 6e 74 69 6f 6e 77 65 0d 6d 75 73 74 20 68 61 76 |ntionwe.must hav| 00001390 65 20 74 68 65 20 62 69 6e 61 72 79 20 70 6f 69 |e the binary poi| 000013a0 6e 74 20 61 74 20 74 68 65 20 68 65 61 64 20 6f |nt at the head o| 000013b0 66 20 74 68 65 20 6e 75 6d 62 65 72 20 73 75 63 |f the number suc| 000013c0 68 0d 74 68 61 74 0d 0d 20 20 20 20 20 20 33 2e |h.that.. 3.| 000013d0 32 35 20 28 64 65 63 69 6d 61 6c 29 20 3d 20 31 |25 (decimal) = 1| 000013e0 31 2e 30 31 20 28 62 69 6e 61 72 79 29 20 3d 20 |1.01 (binary) = | 000013f0 2e 31 31 30 31 20 78 32 5e 32 0d 0d 54 68 69 73 |.1101 x2^2..This| 00001400 20 67 69 76 65 73 20 75 73 20 61 20 72 65 70 72 | gives us a repr| 00001410 65 73 65 6e 74 61 74 69 6f 6e 20 69 6e 20 6d 65 |esentation in me| 00001420 6d 6f 72 79 20 6c 69 6b 65 20 74 68 69 73 0d 0d |mory like this..| 00001430 20 20 45 78 70 6f 6e 65 6e 74 20 20 20 20 20 4d | Exponent M| 00001440 61 6e 74 69 73 73 61 0d 20 20 2d 2d 2d 2d 2d 2d |antissa. ------| 00001450 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------| * 00001480 2d 2d 0d 20 20 7c 20 30 30 30 30 30 30 31 30 20 |--. | 00000010 | 00001490 7c 20 31 31 30 31 30 30 30 30 20 7c 20 30 30 30 || 11010000 | 000| 000014a0 30 30 30 30 30 20 7c 20 30 30 30 30 30 30 30 30 |00000 | 00000000| 000014b0 20 7c 20 30 30 30 30 30 30 30 30 20 7c 0d 20 20 | | 00000000 |. | 000014c0 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------| * 000014f0 2d 2d 2d 2d 2d 2d 2d 2d 0d 0d 73 69 6e 63 65 20 |--------..since | 00001500 74 68 65 20 65 78 70 6f 6e 65 6e 74 20 69 73 20 |the exponent is | 00001510 32 20 28 31 30 20 69 6e 20 62 69 6e 61 72 79 29 |2 (10 in binary)| 00001520 20 61 6e 64 20 74 68 65 20 6d 61 6e 74 69 73 73 | and the mantiss| 00001530 61 20 69 73 0d 61 6c 69 67 6e 65 64 20 74 6f 20 |a is.aligned to | 00001540 74 68 65 20 6c 65 66 74 2e 20 20 49 6e 20 74 68 |the left. In th| 00001550 69 73 20 63 61 73 65 2c 20 6c 69 6b 65 20 6f 6e |is case, like on| 00001560 65 20 6f 66 20 6f 75 72 0d 63 6f 6e 76 65 6e 74 |e of our.convent| 00001570 69 6f 6e 73 20 66 6f 72 20 64 65 63 69 6d 61 6c |ions for decimal| 00001580 20 66 6c 6f 61 74 69 6e 67 20 70 6f 69 6e 74 2c | floating point,| 00001590 20 74 68 65 20 6d 61 6e 74 69 73 73 61 20 69 73 | the mantissa is| 000015a0 0d 61 6c 77 61 79 73 20 62 65 74 77 65 65 6e 20 |.always between | 000015b0 30 20 61 6e 64 20 31 2e 20 20 49 6e 20 65 66 66 |0 and 1. In eff| 000015c0 65 63 74 20 79 6f 75 20 73 68 69 66 74 20 74 68 |ect you shift th| 000015d0 65 20 6d 61 6e 74 69 73 73 61 0d 75 6e 74 69 6c |e mantissa.until| 000015e0 20 69 74 73 20 74 6f 70 20 62 69 74 20 69 73 20 | its top bit is | 000015f0 73 65 74 2e 20 20 54 68 69 73 20 6d 65 61 6e 73 |set. This means| 00001600 20 79 6f 75 20 63 61 6e 20 63 68 65 63 6b 20 74 | you can check t| 00001610 68 65 0d 6e 65 67 61 74 69 76 65 20 66 6c 61 67 |he.negative flag| 00001620 20 28 77 68 69 63 68 20 72 65 66 6c 65 63 74 73 | (which reflects| 00001630 20 62 69 74 20 37 29 20 6f 66 20 74 68 65 20 66 | bit 7) of the f| 00001640 69 72 73 74 20 6d 61 6e 74 69 73 73 61 0d 62 79 |irst mantissa.by| 00001650 74 65 20 69 66 20 79 6f 75 20 77 61 6e 74 20 74 |te if you want t| 00001660 6f 20 63 68 65 63 6b 20 66 6f 72 20 6e 6f 72 6d |o check for norm| 00001670 61 6c 69 73 61 74 69 6f 6e 2e 0d 0d 49 74 20 69 |alisation...It i| 00001680 73 20 76 69 74 61 6c 20 74 6f 20 72 65 6d 65 6d |s vital to remem| 00001690 62 65 72 20 74 68 61 74 2c 20 75 6e 6c 69 6b 65 |ber that, unlike| 000016a0 20 69 6e 74 65 67 65 72 20 6e 75 6d 62 65 72 73 | integer numbers| 000016b0 2c 20 77 68 65 72 65 0d 74 68 65 20 6d 6f 73 74 |, where.the most| 000016c0 20 73 69 67 6e 69 66 69 63 61 6e 74 20 62 79 74 | significant byt| 000016d0 65 20 69 73 20 73 74 6f 72 65 64 20 61 74 20 74 |e is stored at t| 000016e0 68 65 20 67 72 65 61 74 65 73 74 20 61 64 64 72 |he greatest addr| 000016f0 65 73 73 2c 0d 77 69 74 68 20 66 6c 6f 61 74 69 |ess,.with floati| 00001700 6e 67 20 70 6f 69 6e 74 20 69 6e 20 74 68 65 20 |ng point in the | 00001710 42 42 43 20 4d 69 63 72 6f 20 74 68 65 20 65 78 |BBC Micro the ex| 00001720 70 6f 6e 65 6e 74 20 69 73 20 61 74 20 74 68 65 |ponent is at the| 00001730 0d 6c 6f 77 65 73 74 20 61 64 64 72 65 73 73 20 |.lowest address | 00001740 61 6e 64 20 74 68 65 20 6d 61 6e 74 69 73 73 61 |and the mantissa| 00001750 20 69 73 20 73 74 6f 72 65 64 20 69 6e 20 74 68 | is stored in th| 00001760 65 20 6e 65 78 74 20 66 6f 75 72 0d 68 69 67 68 |e next four.high| 00001770 65 72 20 62 79 74 65 73 2c 20 77 69 74 68 20 74 |er bytes, with t| 00001780 68 65 20 6d 6f 73 74 20 73 69 67 6e 69 66 69 63 |he most signific| 00001790 61 6e 74 20 62 79 74 65 20 69 6e 20 74 68 65 20 |ant byte in the | 000017a0 4c 4f 57 45 53 54 0d 61 64 64 72 65 73 73 2e 0d |LOWEST.address..| 000017b0 0d 4e 6f 77 2c 20 62 65 63 61 75 73 65 20 74 68 |.Now, because th| 000017c0 65 20 66 69 72 73 74 20 62 69 74 20 6f 66 20 74 |e first bit of t| 000017d0 68 65 20 6d 61 6e 74 69 73 73 61 20 69 73 20 41 |he mantissa is A| 000017e0 4c 57 41 59 53 20 67 6f 69 6e 67 0d 74 6f 20 62 |LWAYS going.to b| 000017f0 65 20 73 65 74 2c 20 75 6e 6c 65 73 73 20 74 68 |e set, unless th| 00001800 65 20 6e 75 6d 62 65 72 20 69 73 20 7a 65 72 6f |e number is zero| 00001810 2c 20 42 41 53 49 43 20 75 73 65 73 20 74 68 61 |, BASIC uses tha| 00001820 74 20 74 6f 70 0d 62 69 74 20 74 6f 20 64 65 6e |t top.bit to den| 00001830 6f 74 65 20 73 69 67 6e 2e 20 20 54 68 69 73 20 |ote sign. 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But.remember| 000018e0 20 74 68 61 74 20 74 68 69 73 20 69 73 20 75 73 | that this is us| 000018f0 69 6e 67 20 61 20 62 69 74 20 74 68 61 74 20 6f |ing a bit that o| 00001900 75 67 68 74 20 74 6f 20 62 65 20 73 65 74 20 61 |ught to be set a| 00001910 73 20 61 0d 66 6c 61 67 20 61 6e 64 20 69 6e 20 |s a.flag and in | 00001920 66 61 63 74 20 74 68 65 20 74 6f 70 20 62 69 74 |fact the top bit| 00001930 20 69 73 20 61 63 74 75 61 6c 6c 79 20 61 6c 77 | is actually alw| 00001940 61 79 73 20 73 65 74 2e 0d 0d 4f 6e 65 20 66 69 |ays set...One fi| 00001950 6e 61 6c 20 63 6f 6e 74 72 69 76 61 6e 63 65 20 |nal contrivance | 00001960 6f 66 20 42 42 43 20 42 41 53 49 43 20 69 73 20 |of BBC BASIC is | 00001970 74 6f 20 61 64 64 20 26 38 30 20 74 6f 20 74 68 |to add &80 to th| 00001980 65 0d 65 78 70 6f 6e 65 6e 74 20 74 6f 20 61 69 |e.exponent to ai| 00001990 64 20 69 6e 20 63 61 6c 63 75 6c 61 74 69 6e 67 |d in calculating| 000019a0 2c 20 73 6f 20 49 20 68 61 76 65 20 64 6f 6e 65 |, so I have done| 000019b0 20 74 68 61 74 20 62 65 6c 6f 77 2e 0d 0d 48 65 | that below...He| 000019c0 6e 63 65 2c 20 74 68 69 73 20 69 73 20 74 68 65 |nce, this is the| 000019d0 20 72 65 70 72 65 73 65 6e 74 61 74 69 6f 6e 20 | representation | 000019e0 6f 66 20 33 2e 32 35 0d 0d 0d 20 20 45 78 70 6f |of 3.25... 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Exp| 00001b00 6f 6e 65 6e 74 20 20 20 20 20 4d 61 6e 74 69 73 |onent Mantis| 00001b10 73 61 0d 20 20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |sa. -----------| 00001b20 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------| * 00001b40 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 0d 20 20 |-------------. | 00001b50 7c 20 31 30 30 30 30 30 31 30 20 7c 20 31 31 30 || 10000010 | 110| 00001b60 31 30 30 30 30 20 7c 20 30 30 30 30 30 30 30 30 |10000 | 00000000| 00001b70 20 7c 20 30 30 30 30 30 30 30 30 20 7c 20 30 30 | | 00000000 | 00| 00001b80 30 30 30 30 30 30 20 7c 0d 20 20 2d 2d 2d 2d 2d |000000 |. -----| 00001b90 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d 2d |----------------| * 00001bc0 2d 2d 2d 0d 20 20 20 20 26 38 32 20 20 20 20 20 |---. &82 | 00001bd0 20 20 20 26 44 30 20 20 20 20 20 20 20 20 26 30 | &D0 &0| 00001be0 30 20 20 20 20 20 20 20 20 26 30 30 20 20 20 20 |0 &00 | 00001bf0 20 20 20 20 26 30 30 0d 0d 5a 65 72 6f 20 69 73 | &00..Zero is| 00001c00 20 73 74 6f 72 65 64 20 61 73 20 66 69 76 65 20 | stored as five | 00001c10 7a 65 72 6f 20 62 79 74 65 73 2e 0d 0d 49 6e 20 |zero bytes...In | 00001c20 63 61 6c 63 75 6c 61 74 69 6e 67 20 77 69 74 68 |calculating with| 00001c30 20 66 6c 6f 61 74 69 6e 67 20 70 6f 69 6e 74 20 | floating point | 00001c40 6e 75 6d 62 65 72 73 20 6f 66 20 74 68 69 73 20 |numbers of this | 00001c50 66 6f 72 6d 61 74 20 77 65 0d 68 61 76 65 20 74 |format we.have t| 00001c60 68 72 65 65 20 6f 70 65 72 61 74 69 6f 6e 73 2e |hree operations.| 00001c70 20 20 46 69 72 73 74 6c 79 20 77 65 20 63 61 72 | Firstly we car| 00001c80 72 79 20 6f 75 74 20 74 68 65 20 63 61 6c 63 75 |ry out the calcu| 00001c90 6c 61 74 69 6f 6e 0d 6f 6e 20 74 68 65 20 6d 61 |lation.on the ma| 00001ca0 6e 74 69 73 73 61 65 2c 20 73 65 63 6f 6e 64 6c |ntissae, secondl| 00001cb0 79 20 77 65 20 63 61 72 72 79 20 6f 75 74 20 74 |y we carry out t| 00001cc0 68 65 20 6f 70 65 72 61 74 69 6f 6e 20 6f 6e 20 |he operation on | 00001cd0 74 68 65 0d 65 78 70 6f 6e 65 6e 74 73 20 61 6e |the.exponents an| 00001ce0 64 20 66 69 6e 61 6c 6c 79 20 77 65 20 6e 6f 72 |d finally we nor| 00001cf0 6d 61 6c 69 73 65 20 74 68 65 20 72 65 73 75 6c |malise the resul| 00001d00 74 2e 0d 0d 54 68 65 20 70 72 6f 67 72 61 6d 20 |t...The program | 00001d10 69 6e 20 74 68 69 73 20 6d 6f 64 75 6c 65 20 74 |in this module t| 00001d20 61 6b 65 73 20 61 20 70 61 69 72 20 6f 66 20 6e |akes a pair of n| 00001d30 75 6d 62 65 72 73 20 79 6f 75 20 65 6e 74 65 72 |umbers you enter| 00001d40 0d 61 6e 64 20 63 61 6c 63 75 6c 61 74 65 73 20 |.and calculates | 00001d50 74 68 65 20 73 75 6d 20 61 6e 64 20 64 69 66 66 |the sum and diff| 00001d60 65 72 65 6e 63 65 20 6f 66 20 74 68 65 6d 20 75 |erence of them u| 00001d70 73 69 6e 67 20 6d 6f 64 69 66 69 65 64 0d 76 65 |sing modified.ve| 00001d80 72 73 69 6f 6e 73 20 6f 66 20 74 68 65 20 6d 75 |rsions of the mu| 00001d90 6c 74 69 2d 62 79 74 65 20 61 72 69 74 68 6d 65 |lti-byte arithme| 00001da0 74 69 63 61 6c 20 72 6f 75 74 69 6e 65 73 20 49 |tical routines I| 00001db0 20 68 61 76 65 0d 69 6e 74 72 6f 64 75 63 65 64 | have.introduced| 00001dc0 20 65 61 72 6c 69 65 72 20 69 6e 20 74 68 65 20 | earlier in the | 00001dd0 73 65 72 69 65 73 2e 0d 0d 49 20 77 6f 6e 27 74 |series...I won't| 00001de0 20 67 6f 20 69 6e 74 6f 20 67 72 65 61 74 20 64 | go into great d| 00001df0 65 74 61 69 6c 20 68 65 72 65 20 61 62 6f 75 74 |etail here about| 00001e00 20 74 68 65 20 70 72 6f 67 72 61 6d 20 73 69 6e | the program sin| 00001e10 63 65 20 69 74 0d 69 73 20 63 6f 70 69 6f 75 73 |ce it.is copious| 00001e20 6c 79 20 61 6e 6e 6f 74 61 74 65 64 20 62 75 74 |ly annotated but| 00001e30 20 61 20 66 65 77 20 70 6f 69 6e 74 73 20 61 72 | a few points ar| 00001e40 65 20 77 6f 72 74 68 0d 6d 65 6e 74 69 6f 6e 69 |e worth.mentioni| 00001e50 6e 67 2e 0d 0d 54 68 65 20 69 6e 70 75 74 20 6f |ng...The input o| 00001e60 66 20 74 68 65 20 66 6c 6f 61 74 69 6e 67 20 70 |f the floating p| 00001e70 6f 69 6e 74 20 6e 75 6d 62 65 72 73 20 69 73 20 |oint numbers is | 00001e80 64 6f 6e 65 20 75 73 69 6e 67 20 61 0d 70 73 65 |done using a.pse| 00001e90 75 64 6f 20 4f 50 54 20 66 75 6e 63 74 69 6f 6e |udo OPT function| 00001ea0 20 64 65 66 69 6e 65 64 20 61 74 20 74 68 65 20 | defined at the | 00001eb0 65 6e 64 20 6f 66 20 74 68 65 20 70 72 6f 67 72 |end of the progr| 00001ec0 61 6d 2e 20 20 49 74 0d 72 65 6c 69 65 73 20 6f |am. 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In | 000020f0 66 61 63 74 20 74 68 65 20 42 41 53 49 43 20 66 |fact the BASIC f| 00002100 70 20 72 6f 75 74 69 6e 65 73 20 61 6c 73 6f 20 |p routines also | 00002110 75 73 65 20 61 20 72 6f 75 6e 64 69 6e 67 0d 62 |use a rounding.b| 00002120 79 74 65 20 74 6f 20 69 6d 70 72 6f 76 65 20 70 |yte to improve p| 00002130 72 65 63 69 73 69 6f 6e 20 62 75 74 20 74 68 61 |recision but tha| 00002140 74 20 64 6f 65 73 20 6e 6f 74 20 73 65 65 6d 20 |t does not seem | 00002150 74 6f 20 62 65 0d 6e 65 63 65 73 73 61 72 79 20 |to be.necessary | 00002160 68 65 72 65 2e 0d 0d 54 68 65 20 6d 61 69 6e 20 |here...The main | 00002170 72 6f 75 74 69 6e 65 73 20 66 6f 72 20 61 64 64 |routines for add| 00002180 69 74 69 6f 6e 20 61 6e 64 20 73 75 62 74 72 61 |ition and subtra| 00002190 63 74 69 6f 6e 20 61 72 65 0d 73 74 72 61 69 67 |ction are.straig| 000021a0 68 74 66 6f 72 77 61 72 64 20 65 6e 6f 75 67 68 |htforward enough| 000021b0 2e 20 20 54 68 65 79 20 61 72 65 20 62 61 73 69 |. They are basi| 000021c0 63 61 6c 6c 79 20 66 69 76 65 20 62 79 74 65 0d |cally five byte.| 000021d0 6d 75 6c 74 69 70 6c 65 20 70 72 65 63 69 73 69 |multiple precisi| 000021e0 6f 6e 20 73 69 6d 69 6c 61 72 20 74 6f 20 74 68 |on similar to th| 000021f0 6f 73 65 20 69 6e 20 65 61 72 6c 69 65 72 20 6d |ose in earlier m| 00002200 6f 64 75 6c 65 73 2e 20 20 54 68 65 0d 63 75 72 |odules. 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I| 000023d0 74 20 64 6f 65 73 20 74 68 69 73 20 62 79 20 72 |t does this by r| 000023e0 6f 74 61 74 69 6e 67 20 74 68 65 20 6d 61 6e 74 |otating the mant| 000023f0 69 73 73 61 20 72 69 67 68 74 20 28 77 68 69 63 |issa right (whic| 00002400 68 0d 6d 61 6b 65 73 20 69 74 20 73 6d 61 6c 6c |h.makes it small| 00002410 65 72 29 20 77 68 69 6c 65 20 69 6e 63 72 65 61 |er) while increa| 00002420 73 69 6e 67 20 74 68 65 20 65 78 70 6f 6e 65 6e |sing the exponen| 00002430 74 20 75 6e 74 69 6c 20 74 68 65 0d 65 78 70 6f |t until the.expo| 00002440 6e 65 6e 74 73 20 6d 61 74 63 68 2e 0d 0d 27 6e |nents match...'n| 00002450 65 67 5f 63 6f 6e 76 65 72 74 27 20 6d 61 6b 65 |eg_convert' make| 00002460 73 20 61 6e 79 20 6e 65 67 61 74 69 76 65 20 6e |s any negative n| 00002470 75 6d 62 65 72 73 20 69 6e 74 6f 20 32 27 73 0d |umbers into 2's.| 00002480 63 6f 6d 70 6c 65 6d 65 6e 74 2c 20 6d 61 6b 69 |complement, maki| 00002490 6e 67 20 75 73 65 20 6f 66 20 74 68 65 20 65 78 |ng use of the ex| 000024a0 74 72 61 20 6f 76 65 72 66 6c 6f 77 20 62 79 74 |tra overflow byt| 000024b0 65 2e 0d 0d 54 68 65 6e 20 63 6f 6d 65 20 74 68 |e...Then come th| 000024c0 65 20 61 64 64 69 74 69 6f 6e 20 6f 72 20 73 75 |e addition or su| 000024d0 62 74 72 61 63 74 69 6f 6e 20 73 75 62 72 6f 75 |btraction subrou| 000024e0 74 69 6e 65 73 2e 0d 0d 27 6e 65 67 5f 63 6f 6e |tines...'neg_con| 000024f0 76 65 72 74 5f 62 61 63 6b 27 20 74 61 6b 65 73 |vert_back' takes| 00002500 20 61 6e 79 20 72 65 73 75 6c 74 20 74 68 61 74 | any result that| 00002510 20 69 73 20 32 27 73 20 63 6f 6d 70 6c 65 6d 65 | is 2's compleme| 00002520 6e 74 0d 6e 65 67 61 74 69 76 65 20 61 6e 64 20 |nt.negative and | 00002530 6d 61 6b 65 73 20 69 74 20 70 6f 73 69 74 69 76 |makes it positiv| 00002540 65 20 77 68 69 6c 65 20 6d 6f 64 69 66 79 69 6e |e while modifyin| 00002550 67 20 69 74 73 20 73 69 67 6e 0d 62 79 74 65 2e |g its sign.byte.| 00002560 0d 0d 54 68 65 6e 20 77 65 20 74 72 61 70 20 61 |..Then we trap a| 00002570 20 7a 65 72 6f 20 72 65 73 75 6c 74 20 73 69 6e | zero result sin| 00002580 63 65 20 74 68 65 20 7a 65 72 6f 20 66 6f 72 6d |ce the zero form| 00002590 61 74 20 69 73 0d 6e 6f 6e 2d 73 74 61 6e 64 61 |at is.non-standa| 000025a0 72 64 20 61 6e 64 20 77 6f 75 6c 64 20 70 75 74 |rd and would put| 000025b0 20 74 68 65 20 72 65 6e 6f 72 6d 61 6c 69 73 69 | the renormalisi| 000025c0 6e 67 20 72 6f 75 74 69 6e 65 20 69 6e 74 6f 20 |ng routine into | 000025d0 61 6e 0d 65 6e 64 6c 65 73 73 20 6c 6f 6f 70 2e |an.endless loop.| 000025e0 0d 0d 27 72 65 6e 6f 72 6d 61 6c 69 73 65 27 20 |..'renormalise' | 000025f0 61 64 6a 75 73 74 73 20 74 68 65 20 72 65 73 75 |adjusts the resu| 00002600 6c 74 20 75 6e 74 69 6c 20 69 74 20 69 73 20 61 |lt until it is a| 00002610 20 6e 6f 72 6d 61 6c 69 73 65 64 0d 6e 75 6d 62 | normalised.numb| 00002620 65 72 20 28 77 69 74 68 20 74 68 65 20 74 6f 70 |er (with the top| 00002630 20 62 69 74 20 6f 66 20 74 68 65 20 6d 61 6e 74 | bit of the mant| 00002640 69 73 73 61 20 73 65 74 29 20 61 73 20 65 78 70 |issa set) as exp| 00002650 65 63 74 65 64 20 62 79 0d 42 41 53 49 43 2e 20 |ected by.BASIC. | 00002660 20 54 68 65 20 65 78 70 6f 6e 65 6e 74 20 69 73 | The exponent is| 00002670 20 61 64 6a 75 73 74 65 64 20 61 63 63 6f 72 64 | adjusted accord| 00002680 69 6e 67 6c 79 2e 20 20 49 66 20 74 68 65 72 65 |ingly. If there| 00002690 20 69 73 0d 61 6e 79 74 68 69 6e 67 20 69 6e 20 | is.anything in | 000026a0 74 68 65 20 6f 76 65 72 66 6c 6f 77 20 62 79 74 |the overflow byt| 000026b0 65 20 74 68 65 6e 20 74 68 65 20 77 68 6f 6c 65 |e then the whole| 000026c0 20 6d 61 6e 74 69 73 73 61 0d 28 69 6e 63 6c 75 | mantissa.(inclu| 000026d0 64 69 6e 67 20 74 68 65 20 6f 76 65 72 66 6c 6f |ding the overflo| 000026e0 77 20 62 79 74 65 29 20 69 73 20 72 6f 74 61 74 |w byte) is rotat| 000026f0 65 64 20 72 69 67 68 74 20 75 6e 74 69 6c 20 74 |ed right until t| 00002700 68 65 0d 6f 76 65 72 66 6c 6f 77 20 62 79 74 65 |he.overflow byte| 00002710 20 69 73 20 65 6d 70 74 79 2e 20 20 54 68 65 20 | is empty. The | 00002720 65 78 70 6f 6e 65 6e 74 20 69 73 20 69 6e 63 72 |exponent is incr| 00002730 65 61 73 65 64 20 62 79 20 6f 6e 65 0d 66 6f 72 |eased by one.for| 00002740 20 65 61 63 68 20 72 6f 74 61 74 69 6f 6e 2e 20 | each rotation. | 00002750 20 49 66 2c 20 64 75 72 69 6e 67 20 74 68 65 20 | If, during the | 00002760 63 6f 75 72 73 65 20 6f 66 20 74 68 69 73 20 69 |course of this i| 00002770 6e 63 72 65 6d 65 6e 74 2c 0d 74 68 65 20 65 78 |ncrement,.the ex| 00002780 70 6f 6e 65 6e 74 20 62 65 63 6f 6d 65 73 20 7a |ponent becomes z| 00002790 65 72 6f 20 74 68 65 6e 20 77 65 20 68 61 76 65 |ero then we have| 000027a0 20 67 65 6e 65 72 61 74 65 64 20 61 20 6e 75 6d | generated a num| 000027b0 62 65 72 0d 74 6f 6f 20 6c 61 72 67 65 20 66 6f |ber.too large fo| 000027c0 72 20 74 68 65 20 73 79 73 74 65 6d 20 74 6f 20 |r the system to | 000027d0 68 61 6e 64 6c 65 2c 20 61 6e 64 20 61 6e 20 65 |handle, and an e| 000027e0 72 72 6f 72 20 69 73 0d 67 65 6e 65 72 61 74 65 |rror is.generate| 000027f0 64 2e 20 20 49 74 20 68 61 73 20 74 68 65 20 6e |d. It has the n| 00002800 75 6d 62 65 72 20 32 30 20 62 75 74 20 49 20 68 |umber 20 but I h| 00002810 61 76 65 20 67 69 76 65 6e 20 69 74 20 61 0d 64 |ave given it a.d| 00002820 69 66 66 65 72 65 6e 74 20 6d 65 73 73 61 67 65 |ifferent message| 00002830 20 74 6f 20 74 68 65 20 73 74 61 6e 64 61 72 64 | to the standard| 00002840 20 27 74 6f 6f 20 62 69 67 27 2e 0d 0d 49 66 20 | 'too big'...If | 00002850 74 68 65 20 6f 76 65 72 66 6c 6f 77 20 62 79 74 |the overflow byt| 00002860 65 20 69 73 20 65 6d 70 74 79 20 74 68 65 6e 20 |e is empty then | 00002870 74 68 65 20 72 65 73 74 20 6f 66 20 74 68 65 20 |the rest of the | 00002880 6d 61 6e 74 69 73 73 61 0d 69 73 20 72 6f 74 61 |mantissa.is rota| 00002890 74 65 64 20 6c 65 66 74 20 75 6e 74 69 6c 20 74 |ted left until t| 000028a0 68 65 20 74 6f 70 20 62 69 74 20 6f 66 20 74 68 |he top bit of th| 000028b0 65 20 6d 61 6e 74 69 73 73 61 20 69 73 20 73 65 |e mantissa is se| 000028c0 74 2e 20 0d 54 68 65 20 65 78 70 6f 6e 65 6e 74 |t. .The exponent| 000028d0 20 69 73 20 64 65 63 72 65 61 73 65 64 20 62 79 | is decreased by| 000028e0 20 6f 6e 65 20 66 6f 72 20 65 61 63 68 20 72 6f | one for each ro| 000028f0 74 61 74 69 6f 6e 2e 20 20 4f 66 0d 63 6f 75 72 |tation. Of.cour| 00002900 73 65 20 69 66 20 74 68 65 20 74 6f 70 20 62 69 |se if the top bi| 00002910 74 20 69 73 20 61 6c 72 65 61 64 79 20 73 65 74 |t is already set| 00002920 20 74 68 65 6e 20 74 68 65 20 6e 75 6d 62 65 72 | then the number| 00002930 20 69 73 0d 63 6f 72 72 65 63 74 2e 0d 0d 46 69 | is.correct...Fi| 00002940 6e 61 6c 6c 79 20 74 68 65 20 72 6f 75 74 69 6e |nally the routin| 00002950 65 20 61 74 20 27 72 65 70 6c 61 63 65 5f 74 6f |e at 'replace_to| 00002960 70 5f 62 69 74 27 20 63 61 6c 63 75 6c 61 74 65 |p_bit' calculate| 00002970 73 20 74 68 65 20 6e 65 77 0d 73 69 67 6e 20 62 |s the new.sign b| 00002980 69 74 20 61 6e 64 20 72 65 70 6c 61 63 65 73 20 |it and replaces | 00002990 74 68 65 20 74 6f 70 20 62 69 74 20 6f 66 20 74 |the top bit of t| 000029a0 68 65 20 6d 61 6e 74 69 73 73 61 20 77 69 74 68 |he mantissa with| 000029b0 20 69 74 2e 0d 0d 57 68 65 6e 20 79 6f 75 20 72 | it...When you r| 000029c0 75 6e 20 74 68 65 20 70 72 6f 67 72 61 6d 20 79 |un the program y| 000029d0 6f 75 20 6d 61 79 20 68 61 76 65 20 74 6f 20 62 |ou may have to b| 000029e0 65 20 69 6e 20 4d 4f 44 45 20 37 0d 62 65 63 61 |e in MODE 7.beca| 000029f0 75 73 65 20 74 68 65 20 61 73 73 65 6d 62 65 72 |use the assember| 00002a00 20 63 6f 64 65 20 69 73 20 72 61 74 68 65 72 20 | code is rather | 00002a10 6c 6f 6e 67 2e 20 20 20 59 6f 75 20 65 6e 74 65 |long. You ente| 00002a20 72 20 61 0d 63 6f 75 70 6c 65 20 6f 66 20 6e 75 |r a.couple of nu| 00002a30 6d 62 65 72 73 20 6f 66 20 76 69 72 74 75 61 6c |mbers of virtual| 00002a40 6c 79 20 61 6e 79 20 73 69 7a 65 2c 20 61 6e 64 |ly any size, and| 00002a50 20 79 6f 75 20 77 69 6c 6c 20 62 65 0d 61 62 6c | you will be.abl| 00002a60 65 20 74 6f 20 63 6f 6d 70 61 72 65 20 74 68 65 |e to compare the| 00002a70 20 61 64 64 69 74 69 6f 6e 20 61 6e 64 20 73 75 | addition and su| 00002a80 62 74 72 61 63 74 69 6f 6e 20 72 65 73 75 6c 74 |btraction result| 00002a90 73 20 66 72 6f 6d 0d 74 68 69 73 20 6d 61 63 68 |s from.this mach| 00002aa0 69 6e 65 20 63 6f 64 65 20 61 6e 64 20 66 72 6f |ine code and fro| 00002ab0 6d 20 42 41 53 49 43 2e 20 20 54 68 65 20 6d 61 |m BASIC. The ma| 00002ac0 63 68 69 6e 65 20 63 6f 64 65 20 68 65 72 65 20 |chine code here | 00002ad0 69 73 0d 61 6c 6d 6f 73 74 20 74 77 69 63 65 20 |is.almost twice | 00002ae0 61 73 20 66 61 73 74 20 61 73 20 42 41 53 49 43 |as fast as BASIC| 00002af0 20 61 6c 74 68 6f 75 67 68 20 69 6e 20 73 6f 6d | although in som| 00002b00 65 20 63 69 72 63 75 6d 73 74 61 6e 63 65 73 0d |e circumstances.| 00002b10 69 74 20 69 73 20 6e 6f 74 20 71 75 69 74 65 20 |it is not quite | 00002b20 61 73 20 70 72 65 63 69 73 65 2e 20 20 54 68 69 |as precise. Thi| 00002b30 73 20 69 73 20 6f 6e 6c 79 20 6c 69 6b 65 6c 79 |s is only likely| 00002b40 20 74 6f 20 73 68 6f 77 20 61 73 0d 61 20 64 69 | to show as.a di| 00002b50 66 66 65 72 65 6e 63 65 20 6f 66 20 31 20 69 6e |fference of 1 in| 00002b60 20 74 68 65 20 73 6d 61 6c 6c 65 73 74 20 73 69 | the smallest si| 00002b70 67 6e 69 66 69 63 61 6e 74 20 64 69 67 69 74 20 |gnificant digit | 00002b80 62 65 74 77 65 65 6e 0d 74 68 65 20 74 77 6f 20 |between.the two | 00002b90 70 72 6f 66 66 65 72 65 64 20 72 65 73 75 6c 74 |proffered result| 00002ba0 73 2e 0d 0d 4e 65 78 74 20 74 69 6d 65 20 6d 75 |s...Next time mu| 00002bb0 6c 74 69 70 6c 69 63 61 74 69 6f 6e 20 61 6e 64 |ltiplication and| 00002bc0 20 64 69 76 69 73 69 6f 6e 2e 0d | division..| 00002bcb
OS\BITS/T\OSB22.m0
OS\BITS/T\OSB22.m1
OS\BITS/T\OSB22.m2
OS\BITS/T\OSB22.m4
OS\BITS/T\OSB22.m5
.