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04-04-88/T\OSB20

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.

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OSBITS - An Exploration of the BBC Micro at Machine Level

By Programmer

..........................................................


Part 20: Real Numbers


In this module of OSbits I am going to lead you into a
little experimentation.  We will start looking at a section
of computing mathematics that is often ignored by reference
books on the BBC micro .... real numbers.

The terminology gets a little confusing here since I have
seen different reference books give different meanings to
terms dealing with real numbers.  So here are my
definitions.

(i) An INTEGER is a number that is a whole number, in other
words its 'fractional' part is zero.  The number 1234 is an
integer.

(ii) A REAL number is a number that is not a whole number. 
It may have an integer part but it will certainly have a
fractional part.  The number 12.34 is real with an integer
part of 12 and a fractional part of .34 whereas the number
0.456 has no integer part and .456 is its fractional part.
Real numbers are sometimes called 'reals'.

(iii) Fixed point arithmetic is carried out with numbers
which all have fractional parts with the same number of
digits (for example 12.00 and 12.34 and 1.23 and 0.12 if we
have two decimal places).

(iv) Floating point arithmetic is carried out with numbers
which can have any number of digits (including none) in
their fractional part.

These definitions apply equally to numbers with any base. 
The examples above are decimal but we will eventually be
working with binary numbers following those rules.

So far all our arithmetical calculations have been done
using integers.  In these next few modules I will explore
some of the techniques for working with real numbers, first
of all in a fixed point format and then in floating point.

[Although we will not cover it in OSbits, there are methods
for dealing with true fractional quantities using a
computer.  This is where you will genuinely store a quantity
of 1/3 instead of 0.33333333 .... (and so on ad infinitum).]

With machine code we are dealing with bits in memory that
are either on or off, so there is no inherent way of storing
a fractional quantity.  A bit can not be 'half on' for
example.  This is a similar problem to negative numbers, and
so similarly a convention for representing real numbers has
to be found.

There is a convention for floating point representation in 
BBC BASIC which we will come on to in a later module, but
for fixed point we are on our own.  So what are the
considerations?  Don't forget that I will be using decimal
numbers as examples in this module.

The time taken to carry out a multiplication (for example)
depends on the number of digits in the number.  So whereas
we could say that we would store all reals simply by
multiplying by a million (so that 1.2345 became 1234500)
this has a penalty in that most numbers will have lots of
spare zeros which takes up space in memory and slows down
calculating.  The opposite side of this particular coin is
that the greater the number of digits used in such a
representation the more accurately the number is held.  This
is particularly true of fractions that do not have a finite
representation as a 'decimal', like 1/3 or 1/7 if we work in
base 10 and 1/10 if we work in base 2.

With both fixed and floating point arithmetic the trade off
is between accuracy and speed, but in this module let's say
that we are going to use a fixed point decimal format which
always gives four figures after the decimal point.

This means that, for example, 123 becomes 1230000 and
123.456 becomes 1234560.  We have introduced a factor, in
this case it is 10000, to translate our real numbers into a
fixed point format that looks like an integer.  We know that
our numbers are now in fact 10000 times too large and we can
adjust our arithmetic to allow for this.

We must have integer values in order that our machine code
arithmetic routines will work, and so we can take the
numbers multiplied up by our factor and truncated to
integers as the basis of our calculations.  In this way
0.0001234 becomes 1, since the other digits are outside the
range we have set ourselves.  It is from truncations like
this that errors are made.

Let us assume that we have two real numbers A and B and
let's see how the arithmetic now works.  Our 'real' versions
of A and B, multiplied up by our conversion factor Cfac
become Ar and Br.  The number we are looking for on the left
hand side of the equation is the result multiplied by Cfac.

Addition:

      A + B = (Ar + Br) DIV Cfac

therefore, multiplying both sides by Cfac we get

      (A + B) * Cfac = Ar + Br

Similarly for subtraction:

      (A - B) * Cfac = Ar - Br

Multiplication:

      A * B = (Ar * Br) DIV Cfac^2

therefore, multiplying both sides by Cfac we get

      (A * B) * Cfac = (Ar * Br) DIV Cfac

Division:

      A / B =  Ar DIV Br  (the conversion factor cancels)

therefore, multiplying both sides by Cfac we get

      (A / B) * Cfac =  (Ar * Cfac) DIV Br

We have to calculate (Ar * Cfac) DIV Br in the case of
division because a straight Ar DIV Br will produce an
integer answer.  We have to make sure that we have enough
space in the number for the fractional part, and multiplying
by Cfac does that.

In each case the above calculations leave a result that is
too large by the factor Cfac and we remove this factor when
we display the result (by dividing by it) just as we
included it on inputting the original numbers by multiplying
by it.

For working in binary Cfac should be byte sized, an integer
power of 256.  If your whole number is four bytes in size it
could be &100 or &10000 depending on how much fractional
accuracy you want.  In decimal it is easier to work with it
as an integer power of 10 just as here we will work using a
value of 10000 for Cfac.

So what is the point of all this fixed point stuff?  Well
basically it is simpler than floating point, where you have
to keep track of the number of 'decimal' places rather than
it being fixed.  This should speed up calculations.

The program module this week uses this fixed point principle
to carry out some simple calculations.  For simplicity Cfac
is 10000 rather than a byte value.  You can compare the
results between a 'cheap and cheerful' fixed point system
implemented using BASIC integer arithmetic and the proper
floating point maths of BASIC.  I have looped a
multiplication a thousand times so that you can get some
idea of speed.  Assume that the floating point gives the
correct answer and you will see what errors you get from the
fixed point representation.

You will see that with numbers over zero there is little
difference (a tribute to Acorn's programming) but with
smaller numbers the fixed point system becomes over twice as
fast as the floating point one.  I know that the results are
not totally accurate but I think they show that a fixed
point system can be useful where your numbers are generally
fractional.

You will also see that the difference between fixed and
floating speeds is less marked when you enter two different
sized numbers with the smaller one first.  This suggests
that as well as small numbers per se, this fixed point
routine works best when the smaller of the two numbers is
the multiplier rather than the multiplicand.

I have timed multiplying because the project for the next
module involves multiplying, but by adjusting line 160 and
280 you could test other arithmetic.  Next time I will use
fixed point arithmetic to calculate the Mandelbrot set which
is a computer graphic very heavily dependant on
multiplication.

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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|
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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 30 3a 20  |.......Part 20: |
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000000b0  20 4f 53 62 69 74 73 20  49 20 61 6d 20 67 6f 69  | OSbits I am goi|
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000000f0  65 20 77 69 6c 6c 20 73  74 61 72 74 20 6c 6f 6f  |e will start loo|
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000019f0  20 74 72 69 62 75 74 65  20 74 6f 20 41 63 6f 72  | tribute to Acor|
00001a00  6e 27 73 20 70 72 6f 67  72 61 6d 6d 69 6e 67 29  |n's programming)|
00001a10  20 62 75 74 20 77 69 74  68 0d 73 6d 61 6c 6c 65  | but with.smalle|
00001a20  72 20 6e 75 6d 62 65 72  73 20 74 68 65 20 66 69  |r numbers the fi|
00001a30  78 65 64 20 70 6f 69 6e  74 20 73 79 73 74 65 6d  |xed point system|
00001a40  20 62 65 63 6f 6d 65 73  20 6f 76 65 72 20 74 77  | becomes over tw|
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00001a60  68 65 20 66 6c 6f 61 74  69 6e 67 20 70 6f 69 6e  |he floating poin|
00001a70  74 20 6f 6e 65 2e 20 20  49 20 6b 6e 6f 77 20 74  |t one.  I know t|
00001a80  68 61 74 20 74 68 65 20  72 65 73 75 6c 74 73 20  |hat the results |
00001a90  61 72 65 0d 6e 6f 74 20  74 6f 74 61 6c 6c 79 20  |are.not totally |
00001aa0  61 63 63 75 72 61 74 65  20 62 75 74 20 49 20 74  |accurate but I t|
00001ab0  68 69 6e 6b 20 74 68 65  79 20 73 68 6f 77 20 74  |hink they show t|
00001ac0  68 61 74 20 61 20 66 69  78 65 64 0d 70 6f 69 6e  |hat a fixed.poin|
00001ad0  74 20 73 79 73 74 65 6d  20 63 61 6e 20 62 65 20  |t system can be |
00001ae0  75 73 65 66 75 6c 20 77  68 65 72 65 20 79 6f 75  |useful where you|
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00001b10  61 6c 2e 0d 0d 59 6f 75  20 77 69 6c 6c 20 61 6c  |al...You will al|
00001b20  73 6f 20 73 65 65 20 74  68 61 74 20 74 68 65 20  |so see that the |
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00001b40  65 6e 20 66 69 78 65 64  20 61 6e 64 0d 66 6c 6f  |en fixed and.flo|
00001b50  61 74 69 6e 67 20 73 70  65 65 64 73 20 69 73 20  |ating speeds is |
00001b60  6c 65 73 73 20 6d 61 72  6b 65 64 20 77 68 65 6e  |less marked when|
00001b70  20 79 6f 75 20 65 6e 74  65 72 20 74 77 6f 20 64  | you enter two d|
00001b80  69 66 66 65 72 65 6e 74  0d 73 69 7a 65 64 20 6e  |ifferent.sized n|
00001b90  75 6d 62 65 72 73 20 77  69 74 68 20 74 68 65 20  |umbers with the |
00001ba0  73 6d 61 6c 6c 65 72 20  6f 6e 65 20 66 69 72 73  |smaller one firs|
00001bb0  74 2e 20 20 54 68 69 73  20 73 75 67 67 65 73 74  |t.  This suggest|
00001bc0  73 0d 74 68 61 74 20 61  73 20 77 65 6c 6c 20 61  |s.that as well a|
00001bd0  73 20 73 6d 61 6c 6c 20  6e 75 6d 62 65 72 73 20  |s small numbers |
00001be0  70 65 72 20 73 65 2c 20  74 68 69 73 20 66 69 78  |per se, this fix|
00001bf0  65 64 20 70 6f 69 6e 74  0d 72 6f 75 74 69 6e 65  |ed point.routine|
00001c00  20 77 6f 72 6b 73 20 62  65 73 74 20 77 68 65 6e  | works best when|
00001c10  20 74 68 65 20 73 6d 61  6c 6c 65 72 20 6f 66 20  | the smaller of |
00001c20  74 68 65 20 74 77 6f 20  6e 75 6d 62 65 72 73 20  |the two numbers |
00001c30  69 73 0d 74 68 65 20 6d  75 6c 74 69 70 6c 69 65  |is.the multiplie|
00001c40  72 20 72 61 74 68 65 72  20 74 68 61 6e 20 74 68  |r rather than th|
00001c50  65 20 6d 75 6c 74 69 70  6c 69 63 61 6e 64 2e 0d  |e multiplicand..|
00001c60  0d 49 20 68 61 76 65 20  74 69 6d 65 64 20 6d 75  |.I have timed mu|
00001c70  6c 74 69 70 6c 79 69 6e  67 20 62 65 63 61 75 73  |ltiplying becaus|
00001c80  65 20 74 68 65 20 70 72  6f 6a 65 63 74 20 66 6f  |e the project fo|
00001c90  72 20 74 68 65 20 6e 65  78 74 0d 6d 6f 64 75 6c  |r the next.modul|
00001ca0  65 20 69 6e 76 6f 6c 76  65 73 20 6d 75 6c 74 69  |e involves multi|
00001cb0  70 6c 79 69 6e 67 2c 20  62 75 74 20 62 79 20 61  |plying, but by a|
00001cc0  64 6a 75 73 74 69 6e 67  20 6c 69 6e 65 20 31 36  |djusting line 16|
00001cd0  30 20 61 6e 64 0d 32 38  30 20 79 6f 75 20 63 6f  |0 and.280 you co|
00001ce0  75 6c 64 20 74 65 73 74  20 6f 74 68 65 72 20 61  |uld test other a|
00001cf0  72 69 74 68 6d 65 74 69  63 2e 20 20 4e 65 78 74  |rithmetic.  Next|
00001d00  20 74 69 6d 65 20 49 20  77 69 6c 6c 20 75 73 65  | time I will use|
00001d10  0d 66 69 78 65 64 20 70  6f 69 6e 74 20 61 72 69  |.fixed point ari|
00001d20  74 68 6d 65 74 69 63 20  74 6f 20 63 61 6c 63 75  |thmetic to calcu|
00001d30  6c 61 74 65 20 74 68 65  20 4d 61 6e 64 65 6c 62  |late the Mandelb|
00001d40  72 6f 74 20 73 65 74 20  77 68 69 63 68 0d 69 73  |rot set which.is|
00001d50  20 61 20 63 6f 6d 70 75  74 65 72 20 67 72 61 70  | a computer grap|
00001d60  68 69 63 20 76 65 72 79  20 68 65 61 76 69 6c 79  |hic very heavily|
00001d70  20 64 65 70 65 6e 64 61  6e 74 20 6f 6e 0d 6d 75  | dependant on.mu|
00001d80  6c 74 69 70 6c 69 63 61  74 69 6f 6e 2e 0d        |ltiplication..|
00001d8e
04-04-88/T\OSB20.m0
04-04-88/T\OSB20.m1
04-04-88/T\OSB20.m2
04-04-88/T\OSB20.m4
04-04-88/T\OSB20.m5

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