Excess-3

Excess-3 or 3-excess^[1] binary code (often abbreviated as XS-3 or X3^[2][3]) or Stibitz code^[1] (after George Stibitz) is a self complementary binary-coded decimal code and numeral system. It is a biased representation. Excess-3 was used on some older computers as well as in cash registers and hand-held portable electronic calculators of the 1970s, among other uses.

Representation

Biased codes are a way to represent values with a balanced number of positive and negative numbers using a pre-specified number N as a biasing value. Biased codes (and Gray codes) are nonweighted codes. In XS-3, numbers are represented as decimal digits, and each digit is represented by four bits as the digit value plus 3 (the "excess" amount):

• The smallest binary number represents the smallest value (0 − excess).
• The greatest binary number represents the largest value (2^N+1 − excess − 1).

Excess-3 / Stibitz code

Decimal	Excess-3	Stibitz	BCD 8-4-2-1	Binary
−3	0000	pseudo-tetrade	N/A	N/A
−2	0001	pseudo-tetrade	N/A	N/A
−1	0010	pseudo-tetrade	N/A	N/A
0	0011	0011	0000	0000
1	0100	0100	0001	0001
2	0101	0101	0010	0010
3	0110	0110	0011	0011
4	0111	0111	0100	0100
5	1000	1000	0101	0101
6	1001	1001	0110	0110
7	1010	1010	0111	0111
8	1011	1011	1000	1000
9	1100	1100	1001	1001
10	1101	pseudo-tetrade	pseudo-tetrade	1010
11	1110	pseudo-tetrade	pseudo-tetrade	1011
12	1111	pseudo-tetrade	pseudo-tetrade	1100
13	N/A	N/A	pseudo-tetrade	1101
14	N/A	N/A	pseudo-tetrade	1110
15	N/A	N/A	pseudo-tetrade	1111

To encode a number such as 127, one simply encodes each of the decimal digits as above, giving (0100, 0101, 1010).

Excess-3 arithmetic uses different algorithms than normal non-biased BCD or binary positional system numbers. After adding two excess-3 digits, the raw sum is excess-6. For instance, after adding 1 (0100 in excess-3) and 2 (0101 in excess-3), the sum looks like 6 (1001 in excess-3) instead of 3 (0110 in excess-3). In order to correct this problem, after adding two digits, it is necessary to remove the extra bias by subtracting binary 0011 (decimal 3 in unbiased binary) if the resulting digit is less than decimal 10, or subtracting binary 1101 (decimal 13 in unbiased binary) if an overflow (carry) has occurred. (In 4-bit binary, subtracting binary 1101 is equivalent to adding 0011 and vice versa.)

Motivation

The primary advantage of XS-3 coding over non-biased coding is that a decimal number can be nines' complemented (for subtraction) as easily as a binary number can be ones' complemented; just invert all bits. Also, when the sum of two XS-3 digits is greater than 9, the carry bit of a 4-bit adder will be set high. This works because, after adding two digits, an "excess" value of 6 results in the sum. Because a 4-bit integer can only hold values 0 to 15, an excess of 6 means that any sum over 9 will overflow (carry).

Example

BCD to excess-3 converter example (VHDL code):

entity bcdxs3 is
  port (
    a   : in    std_logic;
    b   : in    std_logic;
    c   : in    std_logic;
    d   : in    std_logic;

    an  : inout std_logic;
    bn  : inout std_logic;
    cn  : inout std_logic;
    dn  : inout std_logic;

    w   : out   std_logic;
    x   : out   std_logic;
    y   : out   std_logic;
    z   : out   std_logic
  );
end entity bcdxs3;

architecture dataflow of bcdxs3 is
begin
    an  <=  not a;
    bn  <=  not b;
    cn  <=  not c;
    dn  <=  not d;

    w   <=  (an and b  and d ) or (a  and bn and cn)
         or (an and b  and c  and dn);
    x   <=  (an and bn and d ) or (an and bn and c  and dn)
         or (an and b  and cn and dn) or (a  and bn and cn and d);
    y   <=  (an and cn and dn) or (an and c  and d )
         or (a  and bn and cn and dn);
    z   <=  (an and dn) or (a  and bn and cn and dn);

end architecture dataflow; -- of bcdxs3

See also

• Offset binary, excess-N, biased representation
• Excess-128
• Gray code

References

1. Steinbuch, Karl W.; Weber, Wolfgang (1974) [1967]. Taschenbuch der Informatik - Band II - Struktur und Programmierung von EDV-Systemen. Taschenbuch der Nachrichtenverarbeitung (in German). 2 (3 ed.). Springer Verlag Berlin. pp. 98–100. ISBN 3-540-06241-6.
2. Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, New York, USA: John Wiley & Sons, Inc. p. 11. ISBN 0-471-76180-X. Retrieved 2016-01-03.
3. Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. p. 11. ISBN 0-89874-318-4. Retrieved 2016-01-03. (NB. At least some batches of this reprint edition were misprints with defective pages 115–146.)