Proceedings of the Tomsk Order of the October Revolution and Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 203 1974

INTERPRETING SYSTEM FOR OPERATIONS ON 48-BIT NUMBERS IN FLOATING-POINT MODE ON THE “URAL-11B” DIGITAL COMPUTER

V. K. KIVRAN

(Presented by the scientific seminar of the Department of Computer Engineering, TPI)

To solve many problems on the “Ural-11B” digital computer in floating-point mode, the available 24-bit grid is often insufficient for achieving a specified accuracy of computations. The present work proposes an interpreting system for solving problems with an extended bit grid.

In the proposed system the mantissa of a number occupies 36 bits, and the exponent 12 bits. The range of representable numbers is \(2^{320} \div 2^{-320}\), which corresponds approximately to \(10^{100} \div 10^{-100}\). If the exponent of a number exceeds \(500_8\) in absolute value, the system provides for a stop when the exponent is positive and for representing the number by machine zero when the exponent is negative.

A number in the interpreting system is recorded in two adjacent cells: in the first, the mantissa with its sign; its continuation is in the following incomplete cell, followed by the exponent of the number with its sign. The location of a number in NF is indicated by the address of the full cell containing the high-order bits of the mantissa.

A number is represented in the same form in the pseudo-accumulator, in cells 00040–00043. The second number for the operation is called into cells 00044–00047. The working field of arithmetic operations is 00050–00057.

It should be noted that in programs operating with the IS, one may specify, in the address part of commands, the address of the pseudo-accumulator. The IS makes it possible to execute all arithmetic and logical operations of the instruction system of the “Ural-11B” digital computer, as well as commands for unconditional and conditional transfers of control corresponding to the signals \(\omega_0\), \(\omega_1\), and \(\omega_2\) of operation 01 MA. In addition, the IS makes it possible to execute commands for addressing external devices, provided that the commands do not have a unit in bit 15, i.e., the address part must not exceed 37777. Otherwise, to execute these commands one must exit the IS.

In floating-point mode in the IS system it is possible to perform addition, subtraction, subtraction of moduli, calling a number into the pseudo-accumulator and sending it to an address. In addition, the system provides multiplication and division of numbers, extraction of the square root, normalization of 48-bit numbers (with replacement, after normalization, of the last 12 bits by the exponent of the number), sending zero to the pseudo-accumulator and to an address, conversion of numbers from the decimal system to binary and back, and calls to subroutines.

The indication that an operation is to be performed in floating-point mode is a one in the 15th bit of the instruction, i.e., \(40000_8\) must be added to the address part of the instruction.

Any instruction in floating-point mode may send the result to an address if the instruction is “negative.” For example, the instruction \(41M\ 40000 + A\) adds the numbers in the pseudo-accumulator at address \(A + [M]\) and sends the result to the same address.

In the IS system one may use all modifier counters and all modifiers except the 7th, which may be used only for access to SP with exit from IS.

IS is enabled by the instruction 63701410, after which the next instruction will be executed in the IS system. One may exit IS by the instruction \(23M\ 40000 + A\) to cell \(A + [M]\), or \(63M\ 40000 + A\) to cell \(A\).

The following floating-point operation codes are adopted in IS.

\(00\ M\ 40000 + A\) \((40\ M\ 40000 + A)\) — normalization;
\(01\ M\ 40000 + A\) \((41\ M\ 40000 + A)\) — addition;
\(02\ M\ 40000 + A\) or \(42\ M\ 40000 + A\) — call of a number into the pseudo-accumulator;
\(03\ M\ 40000 + A\) \((43\ M\ 40000 + A)\) — subtraction;
\(04\ M\ 40000 + A\) \((44\ M\ 40000 + A)\) — subtraction of absolute values;
\(05\ M\ 40000 + A\) or \(45\ M\ 40000 + A\) — sending a number to an address;
\(06\ M\ 40000 + A\) \((46\ M\ 40000 + A)\) — multiplication;
\(07\ M\ 40000 + A\) \((47\ M\ 40000 + A)\) — division;
\(10\ M\ 40000 + A^*\) \((50\ M\ 40000 + A)\) — extraction of the square root from the number in the pseudo-accumulator;
\(12\ M\ 40000 + A\) \((52\ M\ 40000 + A)\) — extraction of the square root from the number at the address;
\(13\ M\ 40000 + A^*\) \((53\ M\ 40000 + A)\) — conversion of the number in the pseudo-accumulator from the binary system to the decimal system;
\(14\ M\ 40000 + A\) \((54\ M\ 40000 + A)\) — conversion of the number at the address from the binary system to the decimal system;
\(15\ M\ 40000 + A^*\) \((55\ M\ 40000 + A)\) — conversion of the number in the pseudo-accumulator from the decimal system to the binary system;
\(16\ M\ 40000 + A\) \((56\ M\ 40000 + A)\) — conversion of the number at the address from the decimal system to the binary system;
\(17\ M\ 40000 + A^*\) \((57\ M\ 40000 + A)\) — sending zero to the pseudo-accumulator (or to the pseudo-accumulator and to the address).

Access to SP with exit from IS is carried out by the instruction \(63740000 + A\), where \(A\) is the address of the first SP instruction. In this case it is forbidden to use line-by-line recording of the initial SP information. Therefore, before access to SP, all initial input data must be in the places provided for by SP. The corresponding requirement is also imposed on SP. Thus, in an SP of elementary functions, the argument and the computed value of the function must be located in the pseudo-accumulator.

Access to subprograms operating without exit from IS is performed by the instruction \(63MA\), where \(A\) is the address of the first instruction of the subprogram. Exit from it is carried out, as usual, by the instruction \(23M00000 + 2n\), where \(n\) is the number of instructions skipped after the access. In this case any modifier may be used except the 7th.

The interpreting system IS includes the interpreting program IP for implementing instructions, a block of arithmetic oper—

* — the modifier number and the address \(A\) do not affect the execution of the operation.

of BA operations, the block for converting numbers from the decimal system to binary and back, BP, the library of standard subroutines with the accompanying program BSP, and the arranging program RP for automating work with BSP.

In the present paper, IP, BA, and BP are presented. Their placement in NF must be as follows: in cells 01410÷01754 — IP with constants in 01372÷01406; in cells 00100÷01360 — BA with constants in 01362÷01370; in cells 01754÷03006 — BP with constants in 02422÷02442.

The BA and BP blocks can operate both within the IS system and independently. A description of these blocks is not given, owing to the simplicity of their algorithms and programs.

When BA and BP operate independently, without IS, in order to perform an operation it is necessary to send the numbers for the operation to cells 00040÷00043 and 00044÷00047 and to make the call in the usual way. The result will be located in cells 00040÷00043. In this case it should be borne in mind that the BP block cannot operate without BA. Calls are made to the following cells:

00374 — normalization;
00122 — addition;
00114 — subtraction;
00100 — subtraction of moduli;
00470 — multiplication;
00632 — division;
01070 — extraction of the square root from the number at address
        00040÷00043;
01060 — extraction of the square root from the number at address
        00044÷00047;
01756 — conversion of a number from the binary system to decimal at address
        00044÷00047;
01766 — conversion of a number from the binary system to decimal at address
        00040÷00043;
0244 — conversion of a number from the decimal system to binary at address
        00040÷00043;
02454 — conversion of a number from the decimal system to binary at address
        00044÷00047.

Before performing normalization, the order of the unnormalized number should be sent to the incomplete cell 00056.

When the BA block performs operations, a stop is provided by the command 77700070 if the order of the number is greater than +500₈, upon division by zero, and upon extraction of the square root from a negative number.

Operator scheme of IP

BA                  SP $B_x\downarrow;\ P_1\downarrow P_2\downarrow P_3 Z_1 D P_4\downarrow \downarrow K\downarrow Z_2\downarrow \Phi\downarrow;\ P_5\downarrow P_6P\downarrow;0\downarrow.$ $\quad P_5\qquad P_5\quad \mathrm{SP}\quad -\qquad Z_2\quad \Phi\qquad P_1\quad 0\quad \Phi\quad \Phi$ $\qquad\qquad\qquad \mathrm{Out.}\qquad Z_3\mathrm{BA}$

$B_x$ (01410÷01412) — entry into IS;
$P_1$ (01560÷1562) — separation of commands with floating and fixed point;
$P_2$ (01564÷01576) — separation of commands calling SP from operations with floating point;

$P_3$ (01604÷01610) — check for sending the result to an address;
$Z_1$ (01612÷01632) — call of a number by address;
$D$ (01634÷01646) — command decoder;
$P_4$ (01736÷01754) — operation distributor;
$K$ (01666) — enable writing by address;
$Z_2$ (01670÷01714) — sending the result to an address;
$\Phi$ (01544÷01556) — call of a command into the IS;
$Z_3$ (015554÷01662) — sending a number into the pseudo-adder;
$P_5$ (01600÷01602) — separation of control-transfer commands and execution of fixed-point operations;
(01414÷01442)
$P_6$ (01444÷01524) — decoder of transfer commands;
$\Pi$ (01650÷01652) — transfer of control;
$O$ (01526÷01542) — execution of a fixed-point operation.

The complete program of the interpreting system is given in the appendix.

Appendix

00100	420	00040	00110	520	01362
2	520	01362	2	450	00044
4	450	00040	4	420	00044
6	420	00044	6	540	01364
00120	450	00044	00230	130	00046
2	560	00044	2	600	00466
4	600	00466	4	420	00052
6	560	00040	6	440	01367
00130	600	00142	00240	450	00052
2	420	00043	2	210	00202
4	430	00047	4	020	00046
6	450	00052	6	510	40001
00140	210	00172	00250	050	00046
2	020	00040	2	420	00044
4	050	00050	4	520	01364
6	020	00044	6	530	00046
00150	050	00040	00260	450	00046
2	020	00050	2	420	00042
4	050	00044	4	510	40002
6	020	00042	6	050	00042
00160	050	00050	00270	020	00040
2	020	00046	2	110	00026
4	050	00042	4	530	00042
6	020	00050	6	450	00042
00170	050	00046	00300	020	00040
2	420	00043	2	110	40001
4	410	01367	4	050	00040
6	450	00056	6	020	00046
00200	570	00047	00310	010	00042
2	020	00046	2	050	00042
4	510	40001	4	110	40027
6	050	00046	6	010	00044
00210	420	00045	00320	010	00040
2	510	00013	2	050	00040
4	530	00046	4	200	00334
6	450	00046	6	420	00042

Address	Code	Operand	Address	Code	Operand
00220	020	00044	00330	520	01364
2	110	40001	2	450	00040
4	050	00044	4	540	00042
6	120	01362	6	210	00366
00340	020	00042	00450	450	00043
2	600	00366	2	440	00434
4	120	01364	4	610	00466
6	130	01366	6	560	00043
00350	010	00040	00460	210	01370
2	050	00040	2	170	00040
4	020	00042	4	170	00042
6	120	01362	6	237	00000
00360	140	01362	00470	420	00047
2	010	01366	2	410	00043
4	050	00042	4	450	00056
6	020	00042	6	420	00046
00370	510	00001	00500	510	40001
2	050	00042	2	050	00046
4	020	00040	4	420	00042
6	120	01362	6	450	00052
00400	130	00042	00510	020	00040
2	600	00462	2	050	00050
4	420	00040	4	520	01364
6	520	00410	6	540	00044
00410	200	00446	00520	450	00044
2	020	00040	2	170	00040
4	110	00001	4	170	00042
6	050	00040	6	020	00046
00420	420	00042	00530	110	40001
2	510	40013	2	050	00046
4	530	00041	4	020	00044
6	450	00041	6	110	00026
00430	020	00042	00540	530	00046
2	510	00001	2	450	00046
4	050	00042	4	020	00044
6	420	00056	6	110	40001
00440	430	01367	00550	050	00044
2	450	00056	2	020	00050
4	230	00404	4	510	00001
6	420	00056	6	050	00050
00560	420	00052	00670	520	01362
2	510	40013	2	450	00044
4	530	00051	4	600	01370
6	450	00051	6	420	00633
00570	420	00052	00700	050	00054
2	510	00001	2	020	00040
4	450	00052	4	110	00001
6	130	00050	6	050	00040
00600	600	00626	00710	420	00042
2	560	00050	2	510	40013
4	210	00526	4	530	00041
6	020	00042	6	450	00041
00610	010	00046	00720	020	00042
2	050	00042	2	510	00001
4	110	40027	4	050	00042

6	010	00044	6	560	00057
00620	010	00040	00730	610	00746
2	050	00040	2	020	00052
4	230	00526	4	430	00046
6	020	00040	6	050	00052
00630	230	00324	00740	110	40027
2	420	00043	2	030	00044
4	430	00047	4	230	00760
6	410	01367	6	020	00052
00640	050	00056	00750	410	00046
2	020	00040	2	050	00052
4	120	01362	4	110	40027
6	050	00050	6	010	00044
00650	420	00042	00760	010	00050
2	510	40001	2	410	00055
4	050	00052	4	410	00055
6	420	00044	6	050	00050
00660	540	00040	00770	600	00776
2	450	00040	2	450	00057
4	170	00042	4	230	01006
6	420	00044	6	570	00057
01000	020	00052	01110	610	01370
2	600	01010	2	110	40001
4	450	00057	4	050	00044
6	610	01014	6	420	00043
01010	420	01364	01120	520	01367
2	450	00043	2	600	01156
4	420	00050	4	020	00046
6	520	01002	6	110	40001
01020	450	00055	01130	050	00046
2	020	00050	2	020	00044
4	110	00001	4	110	00026
6	050	00050	6	530	00046
01030	420	00052	01140	450	00046
2	110	40027	2	020	00044
4	010	00050	4	110	40001
6	050	00050	6	050	00044
01040	020	00052	01150	420	00043
2	110	00001	2	410	01367
4	050	00052	4	450	00043
6	420	00054	6	420	00043
01050	430	01367	01160	110	40001
2	450	00054	2	410	01367
4	210	00702	4	450	00056
6	230	00374	6	420	01364
01060	020	00044	01170	510	40003
2	050	00040	2	050	00040
4	020	00046	4	050	00050
6	050	00042	6	170	00042
01070	420	00042	01200	170	00052
2	510	40002	2	420	00053
4	050	00046	4	510	00003
6	020	00040	6	200	00626
01100	110	00026	01210	020	00046
2	530	00046	2	030	00042

Address	Code	Operand	Address	Code	Operand
	450	00046		050	00054
	020	00040		110	40027
01220	010	00044	01330	050	00052
2	030	00040	2	020	00050
4	600	01234	4	110	00026
6	610	01264	6	130	00052
01230	050	00044	01340	050	00052
2	230	01242	2	130	00042
4	560	00054	4	050	00042
6	600	01230	6	020	00050
01240	230	01226	01350	110	40001
2	020	00054	2	050	00050
4	050	00046	4	130	00040
6	020	00042	6	050	00040
01250	010	00052	01360	230	01202
2	050	00042	2	377	77777
4	110	40027	4	400	00000
6	010	00050	6	000	00001
01260	010	00040	01370	770	03700
2	230	01276	2	777	00070
4	020	00042	4	007	77776
6	140	00052	6	000	77776
01270	050	00042	01400	000	00002
2	020	00040	2	300	00004
4	140	00050	4	640	02400
6	050	00040	6	000	01666
01300	020	00044	01410	420	00466
2	110	00001	2	050	01534
4	050	00044	4	420	01534
6	420	00046	6	520	01402
01310	110	40026	01420	540	01422
2	010	00044	2	200	01526
4	050	00044	4	420	01534
6	020	00046	6	520	01356
01320	110	00001	01430	540	01356
2	050	00046	2	600	01526
4	020	00052	4	420	01534
6	110	40001	6	520	01370
01440	550	01404	01550	050	01554
2	600	01526	2	657	01664
4	170	03252	4	020	00000
6	550	01405	6	050	01534
01450	600	03246	01560	520	01403
2	550	01652	2	600	01414
4	200	01472	4	020	01714
6	420	01534	6	050	01666
01460	520	01372	01570	420	01534
2	030	01554	2	520	01371
4	030	01400	4	550	01470
6	050	03252	6	600	01534
01470	230	01506	01600	550	01530
2	420	01534	2	600	01526
4	520	01373	4	020	01534
6	600	03246	6	210	01612

Address	Code	Operand	Address	Code	Operand
01500	510	40015	01610	170	01666
2	530	01532	2	120	01374
4	050	03252	4	530	01604
6	420	01534	6	050	01624
01510	230	03240	01620	010	01400
2	530	01544	2	050	01630
4	050	03252	4	020	00000
6	420	01534	6	050	00044
01520	520	01372	01630	020	00002
2	130	01406	2	050	00046
4	050	01534	4	420	01534
6	420	00056	6	520	01371
01530	110	00013	01640	510	40021
2	010	00040	2	010	01740
4	000	00000	4	050	01646
6	050	00040	6	230	01716
01540	110	40013	01650	020	03252
2	450	00056	2	630	01550
4	020	01554	4	020	00044
6	010	01400	6	050	00040
01660	020	00046	01770	050	00062
2	050	00042	2	570	00060
4	570	00056	4	020	02422
6	230	01544	6	050	00044
01670	020	01534	02000	020	02424
2	120	01374	2	050	00046
4	530	01676	4	560	00043
6	050	01706	6	600	02100
01700	010	01400	02010	610	02024
2	050	01712	2	420	00060
4	020	00040	4	410	02427
6	050	00000	6	450	00060
01710	020	00042	02020	637	00632
2	050	00002	2	230	01774
4	230	01544	4	420	00060
6	230	00374	6	430	02427
01720	230	00122	02030	450	00060
2	230	01654	2	637	00470
4	230	00114	4	020	02422
6	230	00100	6	050	00044
01730	230	01670	02040	020	02424
2	230	00470	2	050	00046
4	230	00632	4	560	00043
6	230	01070	6	600	02100
01740	230	01716	02050	610	02024
2	230	01060	2	020	02430
4	230	01766	4	050	00044
6	230	01756	6	020	02432
01750	230	02444	02060	050	00046
2	230	02454	2	420	00060
4	230	00462	4	410	01367
6	020	00044	6	450	00060
01760	050	00040	02070	637	00632
2	020	00046	2	560	00043

4	050	00042	4	600	02100
6	020	00034	6	210	02052
02100	420	00043	02210	130	00046
2	430	02161	2	050	00046
4	450	00043	4	510	40002
6	420	00042	6	050	00052
02110	510	40001	02220	020	00044
2	450	00042	2	510	40001
4	420	00041	4	050	00044
6	510	00013	6	510	00026
02120	530	00042	02230	530	00052
2	450	00042	2	450	00052
4	020	00040	4	020	00044
6	110	40001	6	510	40002
02130	050	00040	02240	050	00050
2	420	00043	2	420	00053
4	410	01367	4	670	00047
6	450	00043	6	450	00047
02140	200	02106	02250	510	00014
2	420	00042	2	670	00052
4	050	00046	4	670	00046
6	020	00040	6	450	00046
02150	120	01362	02260	540	00046
2	050	00044	2	010	00050
4	420	00040	4	010	00044
6	520	01364	6	050	00044
02160	510	40003	02270	120	00054
2	530	02426	2	130	00040
4	050	00040	4	050	00040
6	170	00042	6	120	00054
02170	420	02435	02300	140	00044
2	050	00054	2	050	00044
4	570	00056	4	420	00046
6	020	00046	6	520	00056
02200	510	40001	02310	530	00042
2	050	00046	2	450	00042
4	420	00045	4	520	00056
6	510	00013	6	540	00046
02320	450	00046	02430	240	00000
2	420	00056	2	000	00004
4	510	40004	4	002	00360
6	450	00056	6	356	32624
02330	420	00055	02440	000	04007
2	510	00010	2	000	10470
4	530	00056	4	020	00040
6	450	00056	6	050	00044
02340	020	00054	02450	020	00042
2	510	40004	2	050	00046
4	050	00054	4	020	00004
6	530	00056	6	050	00062
02350	200	02176	02460	020	00034
2	420	00060	2	050	00064
4	520	01364	4	170	00040
6	510	40003	6	170	00042
02360	530	02426	02470	420	02435

Address	Code	Operand	Address	Code	Operand
2	450	00043	2	050	00054
4	420	00060	4	570	00056
6	440	04427	6	020	00044
02370	600	02374	02500	120	00054
2	610	02406	2	010	00040
4	450	00060	4	050	00040
6	420	00043	6	420	00046
02400	670	02434	02510	520	00056
2	450	00043	2	670	00042
4	230	02364	4	450	00042
6	420	00060	6	540	00042
02410	530	00043	02520	010	00040
2	450	00043	2	050	00040
4	020	00062	4	020	00042
6	050	00034	6	510	40001
02420	237	00000	02530	050	00042
2	255	00574	2	420	00041
4	400	00042	4	510	00013
6	500	00012	6	130	00042
02540	050	00042	02650	520	01364
2	510	40002	2	510	00003
4	050	00052	4	530	00040
6	020	00040	6	450	00040
02550	510	40001	02660	420	00047
2	050	00040	2	510	00003
4	510	00026	4	110	40003
6	530	00052	6	450	00060
02560	450	00052	02670	637	00374
2	020	00040	2	020	02436
4	510	40002	4	050	00044
6	050	00050	6	020	02440
02570	420	00053	02700	050	00046
2	670	00043	2	637	00632
4	450	00043	4	651	00000
6	510	00014	6	420	02776
02600	670	00052	02710	450	02735
2	670	00042	2	420	02703
4	450	00042	4	560	00060
6	540	00042	6	600	02764
02610	010	00050	02720	610	02724
2	010	00040	2	420	02443
4	050	00040	4	450	02751
6	420	00056	6	420	00060
02620	510	40004	02730	441	02434
2	450	00056	2	600	02736
4	420	00055	4	610	02754
6	510	00010	6	450	00060
02630	530	00056	02740	021	02422
2	450	00056	2	050	00044
4	020	00054	4	021	02422
6	510	40004	6	050	00046
02640	050	00054	02750	637	00632
2	530	00056	2	230	02726
4	200	02476	4	641	00006
6	420	00044	6	420	02717

02760	450	02735	03240	520	01373
2	230	02726	2	540	01534
4	020	00062	4	450	01534
6	050	00004	6	020	01534
02770	020	00064	03250	120	01376
2	050	00034	2	010	00000
4	237	00000	4	120	01376
6	000	02754	6	230	01512

REFERENCES

1. Universal automatic digital computing machines “Ural-11.” Programming manual. Part I. Instruction system. Programming. BNTI, 1965.