Summary
The reciproot code calculates the reciprocal of the root using the Taylor-Series expansion around x = 1, that is,
1/sqrt(1+ε) = ∑_{k=0} (-1)ᵏ (2k)! / (2ᵏk)!² εᵏ
or
1 1 1⋅3 1⋅3⋅5
------ = 1 - - ε + --- ε² - ----- ε³ + ...
√(1+ε) 2 2⋅4 2⋅4⋅6
The Manchester/Ferranti only had integer computations, so everything is fixed point, which is tracked manually.
As the above formula only works close to 1, first the number is standardized (which we today call normalized), that is, shifted until it's as close to 1 as possible.
In a second step, to bring it closer to one, an approximation φ of the reciprocal square root is found from a table of 16 entries, which is then factored out.
The remaining value is of the form 1+ε for a sufficiently small ε, so in a third step, the reciprocal root of that value is calculated with 7 members of the above series, using the Horner's method.
Then φ is factored in again, and the resulting reciprocal root is calculated. With the reciprocal root it is easy to calculate the non-reciprocal root. Finally, two index registers are set up to keep track of the binary decimal point.
After calling the routine, one can choose where to put the binary decimal point. This is called understandardizing, as the value isn't standardized as described above. There is a table to assist with that based on assumptions on the range of the input, and where the decimal point should be, by choosing a value called S resp. S', and then using some additional code.
Notation
The Manchester/Ferranti notation is a bit unusual from a modern perspective.
The contents of a memory address are written as [...].
The subscript + stands for "treat as unsigned number", while the subscript ± means "treat as signed number" (which is actually two's complement). The effect of the latter is to do a sign extension on load.
Alan Turing liked to use low-endian notations everywhere (see the introduction in [1]). I kept this for binary numbers (so the most significant bits are to the right), all other numbers are in normal notation.
Numbers are represented by a teletype code, so [/C] comes from / for binary 00000 and C for binary 01110.
New values (after the computation) are marked with a prime sign '. I have not figured out the superscript l or 1 in the link, potentially this was the result of a typist misreading the '.
Programme Sheets
The "programme sheets" (or tapes?) are laid out in the same way as the binary digits on the tube.
20 bits form a horizontal (short) "line" (what we call a word today).
One the tube there are two columns of (short) lines with 20 bits each, in 32 rows. A long line (a double word) is formed from two short lines in consecutive rows. This is the data type for numbers.
On the sheet, the most-significant "address characters" (for the columns) are on the left resp. right side; the least-significant "address character" (for the rows) which is common to both columns is in the middle.
So this program contains 4 columns with address /, E, @ and A.
Annotated Listing
A quickly made disassembly with some ad-hoc notation for the B-lines (index registers) follows.
// VSTA [VS]' = L, A' = 0 ; deposit link
E/ /C/@ std [/C]
@/ GK/A [GK]' = M, M' = 0 ; M = position of MSB of [/C]
A/ /@PO B6' = [/@] ; 78
:/ GKPG B6' = B6 - [GK]
S/ GKPG B6' = B6 - [GK] ; B6 = 78 - 2 * [GK]
I/ /C/C D' = [/C]+
U/ E:IN A' = A + D * [E:][B6]+ ; standard shift table??
½/ /CTA [/C]' = L, A' = 0 ; [/C] now has MSB at bit 39
D/ /C/C D' = [/C]+
R/ J:/N A' = A + D * [J:]+ ; [J:] is probably 2^20 * 1/16
J/ VK/A [VK]' = M, M' = 0 ; [VK] = 0..15 + [D:]
N/ VKQO B7' = [VK]
F/ D:QG B7' = B7 - [D:]
C/ N@UC D' = [N@][B7]+ ; table with 16 overlapping entries of 20 bits
K/ /C/N A' = A + D * [/C]+
T/ VK/A [VK]' = M, M' = 0
Z/ VK/N A' = A + D * [VK]+
L/ VK/N A' = A + D * [VK]+
W/ VK/A [VK]' = M, M' = 0
H/ VKTK A' = 2[VK]±
Y/ MKTA [MK]' = L, A' = 0 ; phi
P/ MK/K D' = [MK]± ; start Horner for [MK]
Q/ A@/J M' = M + [A@] ; 00000 00000 00000 00000 00000 00000 11100 11100
O/ E@/½ A' = A - D * [E@]+ ; 00000 00000 00000 00000 00000 00001 01101 01100
B/ VK/A [VK]' = M, M' = 0
G/ S@/J M' = M + [S@] ; 00000 00000 00000 00000 00000 00000 00111 11100
"/ VK/½ A' = A - D * [VK]+
M/ VK/A [VK]' = M, M' = 0
X/ U@/J M' = M + [U@] ; 00000 00000 00000 00000 00000 00000 00011 00010 = 0.2734375
V/ VK/½ A' = A - D * [VK]+
£/ VK/A [VK]' = M, M' = 0
/E D@/J M' = M + [D@] ; 00000 00000 00000 00000 00000 00000 00000 01010 = 0.3125
EE VK/½ A' = A - D * [VK]+
@E VK/A [VK]' = M, M' = 0
AE J@/J M' = M + [J@] ; 00000 00000 00000 00000 00000 00000 00000 00110 = 0.375
:E VK/½ A' = A - D * [VK]+
SE VK/A [VK]' = M, M' = 0
IE IS/J M' = M + [IS] ; out of the blue, use [IS]? Maybe 0.5?
UE VK/½ A' = A - D * [VK]+
½E VK/A [VK]' = M, M' = 0
DE VK/½ A' = A - D * [VK]+
RE VK/A [VK]' = M, M' = 0
JE N@UJ M' = M + [N@][B7]
NE N@UC D' = [N@][B7]+ ; table
FE VK/F A' = A + D * [VK]±
CE VK/A [VK]' = M, M' = 0
KE BAT: A' = 0 ; also: address = BA
TE US/C D' = [US]+
ZE GK/N A' = A + D * [GK]+
LE GKTA [GK]' = L, A' = 0 ; sets ["K] = \alpha
WE GKQO B7' = [GK]
HE OA/C D' = [OA]+ ; 01011 11111 00110 01100 11110 01000 11010 01101
YE E:/O b-jump [E:]± ; to QE (skip 1)
PE IS/C D' = [IS]+ ; 0.5?
QE VK/N A' = A + D * [VK]+
OE :C/A [:C]' = M, M' = 0 ; recip root
BE :C/C D' = [:C]+
GE E:/O b-jump [E:]± ; to ME (skip 1)
"E /C/N A' = A + D * [/C]+
ME /C/N A' = A + D * [/C]+
XE IC/A [IC]' = M, M' = 0 ; root
VE QAPO B6' = [QA] ; 40
£E KE/P jump [KE]+ ; to GA
/@ C@// 01110 01000 00000 00000 ; 78
E@ //// 00000 00000 00000 00000 ; a6 (taylor coefficient)
@@ /TPI 00000 00001 01101 01100
A@ //// 00000 00000 00000 00000 ; a5 (taylor coefficient)
:@ //UU 00000 00000 11100 11100
S@ //// 00000 00000 00000 00000 ; a4( taylor coefficient)
I@ //MU 00000 00000 00111 11100
U@ //// 00000 00000 00000 00000 ; a3 (taylor coefficient)
½@ //O½ 00000 00000 00011 00010
D@ //// 00000 00000 00000 00000 ; a2 (taylor coefficient)
R@ ///R 00000 00000 00000 01010
J@ //// 00000 00000 00000 00000 ; a1 (taylor coefficient)
N@ ///N 00000 00000 00000 00110
F@ P/DP 01101 00000 10010 01101 ; 1 table to extract phi, 16 entries
C@ IJHY 01100 11010 00101 10101 ; 2
K@ ANEY 11000 00110 10000 10101 ; 3
T@ DMKH 10010 00111 11110 00101 ; 4
Z@ /P£W 00000 01101 11111 11001 ; 5
L@ DHTW 10010 00101 00001 11001 ; 6
W@ EW@W 10000 11001 01000 11001 ; 7
H@ BCYL 10011 01110 10101 01001 ; 8
Y@ L:DL 01001 00100 10010 01001 ; 9
P@ QZXZ 11101 10001 10111 10001 ; 10
Q@ AHLZ 11000 00101 01001 10001 ; 11
O@ BD½Z 10011 10010 00010 10001 ; 12
B@ XTVT 10111 00001 01111 00001 ; 13
G@ A½YT 11000 00010 10101 00001 ; 14
"@ ECNT 10000 01110 00110 00001 ; 15
M@ TE:T 00001 10000 00100 00001 ; 16
X@ //// 00000 00000 00000 00000
V@ //// 00000 00000 00000 00000
£@ //// 00000 00000 00000 00000
/A VSTA [VS]' = L, A' = 0. ; deposit link
EA KAQO B7' = [KA] ; 12
@A /C/J M' = M + [/C]
AA /C/K D' = [/C]±
:A /C/F A' = A + D * [/C]±
SA :C/A [:C]' = M, M' = 0
IA :CTK A' = 2[:C]±
UA :CTA [:C]' = L, A' = 0 ; [:C] := 2 * [/C] ^ 2
½A E:QG B7' = B7 - [E:] ; loop: test B7
DA E:/O b-jump [E:]± ;
RA NS/P jump [NS]+ ; return if not b-cond
JA :C/C D' = [:C]+
NA :C/N A' = A + D * [:C]+
FA :C/A [:C]' = M, M' = 0 ; [:C] = [:C]^2
CA TA/P jump [TA]+ ; to UA
KA N/// 00110 00000 00000 00000
TA UA// 11100 11000 00000 00000
ZA //// 00000 00000 00000 00000
LA //// 00000 00000 00000 00000
WA //// 00000 00000 00000 00000
HA //// 00000 00000 00000 00000
YA //// 00000 00000 00000 00000
PA NE// 00110 10000 00000 00000 ; 44
QA ½E// 00010 10000 00000 00000 ; 40
OA G£NI 01011 11111 00110 01100
BA K@JP 11110 01000 11010 01101
GA "KPG B6' = B6 - ["K] ; start final part, B6 is 40
"A "KPG B6' = B6 - ["K] ; B6 := 40 - 2["K]
MA VKPB [VK]' = B6
XA PAYO B5' = [PA]
VA VKYG B5' = B5 - [VK] ; B5 := 44 - B6 = 4 + 2["K]
£A NS/P jump [NS]+ ; return
Walkthrough
The code relies on constants in other locations, probably all of that was well-known at the time, but as it's not in the listing, I had to guess most of it.
Given the input value in [/C], which must be between 2^20 and 2^40, after completion, [/C] is now shifted, [:C] contains the reciprocal of the square root (shifted), and [IC] contains the square root itself.
The B-line registers B5 and B6 are also used and contain values related to the shift of the result.
In the first line //, the lower part of the accumulator is saved in VS. This is part of the "Schema A" to call a subroutine.
E/ standardizes [/C], and the following code until ½/ is used to shift [/C]. This seems to use a standard table of powers of two at //, since [E:] is probably zero.
From D/ to T/, a value in [J:] (probably 2^20 * 1/16) is multiplied to the shifted [/C], effectively dividing [/C] by 16. This is used as an index in a overlapping table of 20 bits with 16 entries. While 40 bits are loaded, the lower 20 bits seem to be ignored. The value in the table is multiplied to [/C], squared (Z/ to W/), doubled (H/), and then serves as the ε in the series.
P/ to RE apply Horner's method to evaluate the polynomial, with the coefficients in E@ to N@. One of the coefficients is [IS], another seemingly well-known constant which must be 0.5.
JE to CE adjusts the result with the factored-out value, using again the table.
KE to LE sets up ["K] with what the what is called α in the source text (the shift of the result).
WE to XE make the even-odd adjustments that are also mentioned in the source text, while storing the final computer reciprocal root in [:C], and the root itself in [IC].
VE to £E loads B6 with 40, and then continues at GA.
GA to "A set up B6 as described in the source text, while MA to VA do the same for B5.
Finally, £A returns to the called (apparently the return address is stored by convetion in [NS]).
Interstingly, /A to TA seem to contain a completely unrelated routine that calculates a large power of [/C] and return the result in [:C]. One can see the same "calling convetions" as in the other routine.
Additional sources
There is also a description of the routine in [1], page 80, under "standard routines", with the name RECROOT (and not RECIPROOT):
The routines for mathematical functions were mainly chosen for simplicity of programme and generality of application [rather] than for speed.
[...]
RECROOT.
This was a relatively slow method for the reciprocal square root. It
depends on the case p = 2 of the recurrence relation
u_{n+1} = u_n (1 − 1/p * a * u_n ^ p)
for a^(-1/p). It was necessary for the argument to be within a relatively restricted
range, and for u_0 to satisfy 0.5 < a * (u_0)^2 < 2.
It was often found more convenient to use the logarithm and
exponential, although much slower.
Conclusions
- While the routine uses a lookup table and then does some calculations, there the similarity to the Quake fast inverse square root ends: Mathematically, the approaches are completely different (Taylor-series expansion vs. Newton iteration).
- They new that this method was slow, but used it for easy of programming (as a sort of "proof of concept"?)
- The trick to divide by a constant by multiplying with a different constant and then throwing away the lower part was also used back then
- They already had subroutine calling conventions in place
Literature
[1] Alan Turing’s Manual for the Ferranti Mk. I, manuscript. Digitized version.
[2] Programmers' Handbook (2nd Edition) for the Manchester Electronic Computer, revised by R.A. Brooker, October 1952. First chapter
A=A-D*Sinstruction, encoded as/½. Likewise the linked document mentions the use of B5 and B6, which are only present with the Ferranti Computer. – Raffzahn Feb 23 '23 at 23:37