An Autocode programme consists of an ordered sequence of instructions and other items of information. Each of these is written on a separate line and employs the following symbols:
a b c d e f g h u v w x y z π
i j k l m n o p q r s t
. 0 1 2 3 4 5 6 7 8 9
+ - ≠ = > ≥ * ( , ) » ® / Æ ' ?
Tbe programme is ultimately presented to the machine in the form of a length of perforated paper tape which is scanned by a Photoelectric tape reader, the input unit of the machine. The programme tape is prepared by means of a manual keyboard perforator on which are engraved the standard symbols. The material is punched in the conventional fashion, namely from left to right and down the column. Each instruction is followed by two special symbols CR (carriage return) and LF (line feed) which are provided for this purpose. There is also an erase symbol # which is used for overpunching mistakes.
The symbol on the Ferranti teletypes was a large asterisk which no longer appears in modern fonts. We have substituted it with #.
The instructions read from the tape are placed in the instruction store of the machine; the numerical quantities to which they refer are kept in the number store of the machine. The programme will include instructions to set the initial values of such quantities, either directly or by reading them from a further data tape by a process similar to that by which the instructions themselves were read into the machine.
The instructions fall into two classes, the arithmetical instructions which perform the calculation proper, and the control instructions which organise the calculation (e.g., arrange to repeat cycles of arithmetic, select aiternative courses of action, or, as already mentioned, read further numerical data into the machine). This latter class of instructions are the characteristic features of automatic calculating machines which distinguish them from desk machines which are controlled by the operator himself. Both kinds of instruction need to refer to the working store so that it is appropriate to start by describing the notation used to refer to the numbers stored therein.
The numbers recorded in the working storeare also of two kinds, general variables and indices, which, like the two kinds of instructions, relate mainly to the calculation proper and its organisation. Thus the variables have numerical values in the range 10-70 < X < 1070 and are recorded to a precision of eight/nine decimals, while the indices are restricted to integral values in the range -512 ≤ i ≤511.
These are divided into three sets as follows:
There are 480 main variables which can be divided into a maximum of 15 groups associated with the variable letters:
a b c d e f g h u v w x y z π
For example they can be arranged as a single group of 480 variables v0, v1, .. v479 by writing the directive:
v ® 479
Alternatively they could be arranged in three equal groups: a0, a1, .. a159, b0, b1, .. b159, c0, c1, .. c159. The necessary directive being:
a ® 159 b ® 159 c ® 159
It is intended that these groups shall reflect any natural grouping of the quantities occurring in the problem, and provided that the total number of variables does not exceed 480 the number and size of the groups is at the disposal of the programmer.
In addition to the main variables, there are fifteen special variables represented by the letters:
a b c d e f g h u v w x y z π
employed without a suffix. These will be a common feature to every programme which cares to use them. The special variable π may be assumed to have the value 3.14159... until otherwise altered.
Finally there are l4 primed special variables denoted by:
a' b' c' d' e' f' g' h' u' v' w' x' y' z' (note there is no π')
All the above quantities are kept in the working part of the number store. An auxiliary store normally provides up to 10,752 further variables, but these are less easily accessible and will not be introduced at this stage, since the working store will be sufficient for many applications.
Indices are represented by the 12 letters:
i j k l m n o p q r s t
Although permitted integral values in the range -512 ≤ i ≤511, emphasis is placed on positive values because they are primarily intended to be combined with variables in the form:
xi or x(n-1) or x(s-50)
to represent a free suffix; that is, these expressions may represent any one of the variables x0, x1, x2, ... depending on the particular value of the index in question. Thus if n = 4, then x(n-1) will refer to x3. The last two expressions illustrate the most general form which a suffix may take, namely, (index ± integer). Whatever form it takes, however, the computed value of a suffix must lie within sensible limits. In calculations of a repetitive nature an index will assume a range of values, and to arrange this it is necessary to be able to compute with indices as separate items in much the same way as variables.
In preparing the input tape, all expressions are recorded in a one-dimensional form, thus x3, xi and x(s+50) appear as x3, xi, and x(s+50) respectively. Consequently it is not possible to distinguish, in a product, between (say) xi meaning xi and xi meaning x times i. In order to resolve this difficulty, a convention will be introduced later for ordering the factors in a product.
Explicit numerical values will have to be introduced into the programme at some stage, so that it is necessary to explain how these are written. The standard form is:
integral_part decimal_point fractional_part
omitting what is unnecessary. Thus, as is the case in writing suffices, the decimal point can be omitted in whole numbers. However, absolute standardisation of form is not necessary, so that fifteen may be written as
15.0 15 15. 015.0
All these and similar variations will be accepted by the machine. Similarly the square root of 2 to six significant figures may be written as:
1.41421 or 01.41421 or 001.4142100 etc
and 2/3 to the same precision as:
0.666667 or .666667 or 000.666667000 etc
Any number of digits are allowed in the integral part and up to 24 in the fractional part. (In all cases only the first 9 or 10 significant figures are relevant because inside the machine numbers are restricted to a precision of 29 binary digits.)
The basic form of the instructions for computing variables may be illustrated by the following example:
y = 2mn a(m+1) + amn + man + 0.01m + 0.01n
which gives the new value of the variable to be altered (in this case y) in terms of other quantities.
In general the right hand side may involve any number of products which may each have any number of factors either variables, indices, or constants. As already mentioned it is necessary to distinguish in the one-dimensional form:
y = 2mn a(m+1) + amn + man + 0.01m + 0.01n
between am meaning am and a × m. The convention adopted is that an index immediately following a variable letter is treated as a suffix so that the above expression is interpreted as:
y = 2 × m × n × a(m+1) + (am × n) + (m × an) + (0.01 × m) + (0.01 × n)
As a consequence of these rules, numerical factors will usually be placed at the beginning of a product.
Further examples of instructions in this general class are:
a = 0 xk = xk + 1 xn = x0 + nh x = 1
Products can also be divided by a single quantity:
u = x/a + y/b + z/e y = 2πu/n
are possible instructions.
It is recommended that π1, π2,... be used as temporary working space, e.g, where a complex algebraic expression requires several instructions, the intermediate answers are denoted by π1, π2, etc.
In formulating the right-hand side. of an arithmetical expression it is recommended that single terms be written last, for example:
y = u v w + ab + f + g
as in this form the machine will take less time to evaluate the answer. The basic form of the instructions for computing indices may be illustrated by the following example:
i = 2mn + m + n + 1
The only differences between this and the previous class of instruction are that the quantities on the right hand side are restricted to indices or whole numbers, and that the use of the solidus is not permitted. Further examples of instructions in this class are:
i = 0 n = n + 1 r = 10 p + q
The following instructions are a means of introducing certain elementary functions into the programme. Here the left-hand side value y stands for any variable, and the argument x for any right-hand side expression.
y = Æ sqrt(x) y = Æ sin(x) y = Æ cos(x) y = Æ tan(x) y = Æ exp(x) y = Æ log(x) to base e y = Æ mod(x) modulus of y = Æ int pt(x) integral part y = Æ fr pt(x) fractional part y = Æ sign(x) y= 1 if x ≥ 0 otherwise -1
Examples of instructions in this class are:
u = Æ cos(lx/a + my/b + nz/c) a = Æ log(xx + yy) w = Æ sqrt(xxx) xn = Æ cos(nd)
The following class of instructions involves functions of two variables, which may each be replaced by right-hand side expressions.
z = Æ divide(x,y) that is x/y y = Æ arctan(x,y) that is arctan(y/x) ( -π < z < π the quadrant being determined as if x,y were proportional to cos(z), sin(z) respectively) y = Æ radius(x,y) that is sqrt(x2 + y2)
Examples of instructions in this class are
z = Æ divide(x+y,x-y) u = Æ arctan(aa - bb, 2 ab) ai = Æ radius(xi,yi)
Finally there are the instructions:
i = Æ int pt(x) for converting a variable to an index y = Æ poly(x) a0, n for calculating y= a0 + a1x +...+ anxn The parameter a0 is any first member of a group and n is any index or whole number but not an expression y = Æ parity(n) y=(-1)n where n is any index expression i = Æ max(x0,n,m) for determining the index of the maximum (minimum) element of the set i = Æ min(x0,n,m) xn,...xm Here i denotes any index, x0 any first member, and n, m are indices or whole numbers. If there is no unique maximum (or minimum) element, then that with the least index value is taken.
These are the instructions which organise the calculation and for this reason are sometimes known as the red tape.
The most important instructions in this class are the jump instructions. Arithmetical instructions are normally obeyed in the order in which they are listed, but from time to time it is necessary to select alternative courses of action as in the following sequence of instructions for calculating f(x) where its value is ex when x < 0 and has the value 1+sin(x) when x ≥ 0:
........ jump 1, x ≥ 0 f = Æ exp(x) jump 2 1) f = Æ sin(x) f = 1 + f 2) ......
In this example the first instruction is a conditional jump, i.e., if the condition (in this case x ≥ 0) is satisfied then control jumps to the instruction labelled 1) and then continues to obey instructions from there onwards; otherwise, if the condition is not satisfied, then the next instruction is obeyed in the usual way. Any instruction can be labelled in this way with an integer in the range 1 - 127 inclusive.
The second jump instruction in the above example is an unconditional jump and needs no further explanation. The general form of a conditional jump instruction is:
jump n, α ≥ β (or ≠ = >)
where n is a specific label and α, β are the quantities being compared. These must be either both variables (including a numerical constant) or both indices (including a numerical integer). In this case a minus sign can be written before a constant if it is negative, but elsewhere numbers are treated as essentially positive and the + and - signs are treated as operators. It is not possible to compare a variable directly with an index without first converting the index to variable form. Examples of conditional jump instructions are:
jump l,x ≥ y jump 8,1.41421 > a jump 97,r=s jump 50, i ≠ 2
Associated with the above are the instructions:
jump (n) n) = 3) n) = m)
The instruction n) = 3) makes a subsequent jump (n) equivalent to jump 3. One use for this will be to mark the point of return when calling in a subsequence, thus:
main sequence
........
n) = 3)
jump 10
3) ........
........
n) = 7)
jump 10
7) ........
subsequence
10) .........
.........
jump (n)
In the same way the instruction n) = m) makes a jump (n) instruction transfer control to one of several different points depending on the computed value of m. This device is known as a multi-way switch. When an index is used to remember a label in this way it cannot at the same time for other purposes.
The execution time of n) = m) is very much longer (18 millisecs) than that of n) = 3) (120 microsecs).
There is a special form of conditional jump check(x, y, e, 3), where x, y, e are each any variable or constant, and 3 is any label in the same chapter. This has the effect jump to the instruction labelled 3 if e > abs(x - y), otherwise continue with the next instruction. It can thus be use to compare the computed value x with the expected value y to an absolute accuracy of e. In the case of failure. the value of x could be printed. This instruction may be found useful when developing a new programme.
Two special instructions are provided to simplify cycles of operations. These take the form:
i = p(q)r ........... repeat i = p(-q)r ........... repeat
and arrange to execute the intervening instructions for values of i running from p by increments of q (or -q) to r. Any index may be used in place of i and p, q, r, may be any indices or positive integers, subject of course to the restriction that r-p Is a multiple of q, otherwise the cycle will continue indefinitely.
Each instruction of a pair is associated automatically one with the other by the computer during the initial input of the programme. The process of association will permit the use of instruction pairs entirely within other instruction pairs to a total depth of 8. It is not possible, however, to interleave pairs nor to duplicate either member of a pair, i.e. there must be a one to one correspondence of the opening and closing instructions of each loop, for example:
i = 3(1)5
j = 4(2)6
k = 5(1)7
l = 8(-1)6
m = 1(1)9
repeat
repeat
repeat
repeat
j = 2(1)3
k = 5(1)7
repeat
repeat
repeat
The i cycle has a cycle to depth 4 and a cycle to depth 2 inside it. This example is typical and illustrates the type of flow arrangement which is permissible.
The following sequence:
x = 0 i = 2(1)n p = i - 1 j = 1(1)p q = np + j - 1 x = x + aq repeat repeat
for example, illustrates a cycle within a cycle (for calculating the sum of the super diagonal elements of a square matrix). The element ai j is defined as the element n(i-1) + (j-1) of the sequence of variables with name a
Note that a cycle is always obeyed at least once and that if i, q, or r are written on the left hand side of an instruction in the loop the operation of the loop may be affected. At the end of the last repetition the values of i and r are equal.
The single word instruction end is used to terminate a calculation.
Instructions are provided to read numerical information from the input tape into the working store. These are:
read (x) read (i)
which mean read the next number on the tape and set the specified variable (or index) to this value. As each number is read the tape is advanced to bring the next number to the reading station. Numbers must therefore be punched in the order in which they are required. Each number is written in the manner already described (preceded by minus sign if negative) and when punched must be terminated by CR LF or a double space SP SP. The read instruction will also accept numbers punched in the floating decimal form, for- example, -2.5,-3 (i.e., -2.5 × 10-3 or -0.0025). The two component numbers can be punched in any acceptable form except that now the terminal combination of the first is replaced by the comma. The exponent b takes integer values in the range abs(b) < 127. Further examples of data punched in floating decimal form are:
1,-100 i.e., 10-100 -2.5,1 i.e., -25 1000,-3.0 i.e., 1
The last number being in non-standard form. In all cases it is necessary to check that the size of the resulting product does not exceed the capacity of the machine, i.e., abs(a.10b) < 2256.
Note: the floating decimal form applies only to numbers on a data tape, it must not be used for constants appearing in the programme.
To print results the simplest procedure is to write a ? symbol before or after the arithmetical instructions giving the relevant value of the quantity in question, for example:
x = zyy - l ? i = i + 1 ?
This will cause the new value of x (or i) to be printed immediately after computation. Each number is printed on a new line to 10 significant figures, so that results obtained in this fashion will be listed in a single column at the left hand margin of the page.
In case the results are required in tabular form, the following instructions are provided:
print (x) m, n space newline
The first instruction prints the numerical value of x (any variable expression) in fixed decimal point style with m, n positions allowed respectively for the integral and fractional parts. The parameters m, n will usually be preassigned positive integers, for example:
print(2/r) 1, 5
but they may also be indices. The numerical value of x is rounded off by adding 0.5 × 10-n. Integers are treated precisely. If n = 0 the decimal point is omitted. One figure is always printed before the decimal point but other non-significant zeros are suppressed. If negative a minus sign appears before the first digit printed, otherwise a space. If numbers exceed about 1010 they will be printed in floating decimal form (See next paragraph).
The print instruction will print numbers in standard floating decimal form on setting m=0. Three characters (after the comma) are allowed for the exponent - corresponding to a number printed with m=2, n=0. Tbe true zero of the machine, namely 0.2-256, is printed as 0,-128 which will be interpreted correctly if subsequently read by the read instruction in spite of the fact that the exponent -128 lies outside the permissible range. (If a=0 the number (a, b) is treated as 0.2-256 whatever the value of b.)
The query (?) print 10 is equivalent to a newline followed by a formal print with m=0 and n=10 for variables or m=1, n=0 for indices.
Each number is automatically followed by two spaces but extra spaces can be programmed by means of the space instruction. A standard teleprinter carriage allows for up to 68 characters across the page. The instruction newline is equivalent to the operations of line feed and carriage return on a teleprinter.
All numbers printed by the machine can subsequently read in again by means of the read instruction, i.e. input and output are complementary.
Tabulate Sievert's integral ∫
y
0 ea sec x dx
for y=1(1)90° and particular positive values of a.
The method adopted is to tabulate the integrand for x= 0(0.5) 90° and then
calculate the integral step-by-step using Simpson's rule to evaluate the increments; thus
∫
y+h
0 = ∫
y
0 + h/6 [f(y) + 4 f(y+0.5h) + f(y+h)]
where h=π/360.
f ® 180
8 h = π/360 (5ms)
2 s = 0(1)179 1
10 b = Æ cos(sh) (23ms)
12 fs = Æ exp(-a/b) (23ms)
4 repeat 4
4 f180 = 0 4
4 y = 0 4
2 r = 1(1)90 1
4 newline (90ms)
7 print (r) 2,0 (150ms)
2 space (60ms)
11 s = 2r - 1 19
10 f = f(s-1) + 4fs + f(s+1) 22
9 y = y + hf/3 (5ms)
7 print(y) 1,6 (333ms)
4 repeat 4
end
The case s = 180 is excluded from the first loop in order to avoid forming -a/b for b = O. Otherwise the programme itself needs no explanation. The numbers on the left give the number of registers occupied by each instruction in the instruction store, while those on the right give the execution time in units of 60 µs., unless otherwise stated. Most functions take approximately 23 ms., unless treated as quickies (see Part 2), in which case the time is reduced to about 6 ms. Approximate rules for estimating the space and time requirements are given at the end of Part 2.
For technical reasons, large programmes have to be partitioned into chapters, each chapter being restricted in length to 832 registers. (The above programme is well within this limit and would thus constitute a single chapter.) Each chapter has its own labelling system, and to jump from one chapter to another an across instruction is employed, for example:
across 2/3
which means jump to the instruction labelled 2 in chapter 3. Since this is a comparatively time consuming instruction (180 ms.) it should not be included within an inner loop and as far as possible partition into chapters should correspond to distinct parts of the calculation.
chapter 1 directives instructions close chapter 2 directives instructions close chapter 3 directives instructions close
This illustrates the layout of a multi-chapter programme. Each chapter is headed with the chapter number and the variable setting directives, and terminates with the directive word close. If the directives in chapter 3 are the same as those in chapter 2 then we may simply write:
chapter 3 variables 2 instructions close
While if extra directives are to be added we may write:
chapter 3
variables 2
x ® 99
close
Note however that this substitutional directive may only refer to a previous chapter on the programme tape. It is necessary to arrange chapters in ascending numerical order as shown.
The associated instructions:
i = p(q)r ......... repeat l) = 3) ......... jump (l) l) = m)
must occur in the same chapter although intervening instructions may temporarily switch control to another chapter.
(Normally the indices 'l' and 'o' should be used as little as possible since they are readily confused with 1 and 0. In the above case, however. a connection exists between 1 and label, and there is little risk of confusion.)
This means print space available and is intended to be inserted between the last instruction of a chapter and the close directive. The effect Is to print on a newline, the chapter number followed by the number of unused registers at the end of the chapter. The directive may also be used at any point in the chapter so as to record how the chapter space is used up.
When resetting directives in subsequent chapters it may be necessary to appreciate the significance of these statements. Thus, for example:
chapter 1 a ® 99 allocates storage locations 0 to 99 to a0 onwards b ® 99 allocates storage locations 100 to 199 to b0 onwards c ® 99 allocates storage locations 200 to 299 to c0 onwards chapter 2 c ® 99 allocates storage locations 0 to 99 to c0 onwards x ® 49 allocates storage locations 100 to 149 to x0 onwards y ® 49 allocates storage locations 150 to 199 to y0 onwards
Thus the cs of chapter 1 are not those of chapter 2. If it is intended that they should be, then the latter directives might be recast as follows:
x ® 49 y ® 49 π ® 99 (waste) c ® 99
A further consequence of this scheme is that x50 is identical with y0 , x51 is identical with y1 and so on. Suoh overlapping references are sometimes useful. However one must not try to refer to x-1 since there are no variables preceding those defined by the x directive.
At any point within a chapter it is possible to call in a subchapter and subsequently to return to the original chapter at the instruction following the point of departure. This is done by means of a down instruction in the main chapter and an up instruction in the subchapter. For example:
down 2/3
calls in chapter 3 as a subchapter and enters it at the instruction labelled 2. When the subchapter has completed its task the single word instruction:
up
will return control to the main chapter at the instruction following the original down instruction. Alternatively the up instruction may be used in any chapter reached by means of across instructions from the original subchapter. A subchapter may have its own sub-subchapter, but there the regression stops.
The special variable π is reset to 3.14159... at every chapter change, i.e. as a result of the instructions across, down, up; and also (see later) preserve and restore.
Following the last chapter on the programme tape is chapter 0. This is constituted in the same way as any other chapter. but the close directive terminates the programme input process and initiates the programme itself at the first instruction (whether labelled or not) of chapter 0. If there is only one chapter in the programme this could be chapter 0, but it is more usual to regard chapter 0 as a steering chapter. The two alternatives are illustrated below where the calculation of Sievert's Integral, given earlier, is arranged for values of the parameter a = 1(0.25)5.
chapter 0
f ® 180
n = 4(1)20
a = 0.25n
programme proper
repeat
end
close
and as two chapters:
chapter 1
f ® 180
1) programme proper
up
close
chapter 0
n = 4(1)20
a = 0.25n
down 1/1
repeat
end
close
It has been stated above that a programme consists of chapters numbered 1, 2, 3.....,0, and that when chapter 0 has been read the machine stops reading programme and starts obeying it. At this time the number store and any part of the instruction store not filled with instructions are in a standard state.
However. there is a special instruction rmp (read more programme) whose execution makes the machine resume reading programme. Thus chapter 0 can be followed either by a completely new programme which will entirely supersede the old one, or by a new chapter 0 that will supersede the old chapter 0. Thus a succession of steering chapter 0's may be written. In either case after reading another chapter 0 the machine again stops reading programme and starts obeying it. Whereas when execution of the original programme started the number store was in a standard state, the new (or modified) programme finds the numbers left in the number store by the previous programme.
When a calculation has to be repeated for miscellaneous values of certain parameters, it is sometimes convenient to initiate each individual case by restarting with a supplementary chapter 0, which sets up the parameters in question before entering the programme proper. Such a succession of chapter 0's is called a steering programme. The machine can proceed to successive cases automatically by terminating the programme cycle with the rmp instruction. Since numerical results are unaffected by restarting, the successive chapter 0' s need only refer to those parameters which are altered in passing from one case to the next. However this may rule out the possibility of initiating the individual cases out of sequence, or of rescuing the calculation in the event of machine breakdown.
For this reason it is preferable to make the individual chapter 0's independent by resetting if necessary all initial data. If there is a large mass of data, such as a table, to be generated, and which is common to each case, then this should be initiated by the main programme and recorded for future reference in an appropriate part of the store. If this part of the programme also terminates with rmp, the machine can proceed automatically to read the first chapter 0 of the steering programme. If the calculations have to be spread over more than one machine session. or in the event of serious machine breakdown. the main programme will have to be read again. If the programme bas been suitably written, this can immediately be followed by any chapter 0 on the steering tape. The following diagram illustrates the arrangement of programme for three "cases" of a calculation specified by two parameters.
main programme chapter 1 directives 1) main calculation rmp close chapter 0 variables 1 initial calculation rmp close
steering programme chapter 0 variables 1 n = 1 c = 2.8 across 1/1 close chapter 0 variables 1 n = 2 c = 2.9 across 1/1 close chapter 0 variables 1 n = 2 c = 3.0 across 1/1 close
The instruction hoot interrupts the calculation to give a single note (1 kc/s) of duration approximately 1 sec on the loudspeaker. This may prove useful when planning a production run with a steering tape, by arranging to hoot as each case is finished, the operator is thereby reminded that the machine is behaving correctly, and should it jump out of control the interruption would be noticed. In this event the operator can restart the last case manually.