Excess-3

Excess-3 binary (XS-3) or Stibitz code, also called biased representation or Excess-N, is a complementary BCD code and numeral system. Excess-3 was used on some older computers as well as in cash registers and hand held portable electronic calculators of the 1970's, among other uses. It is a way to represent values with a balanced number of positive and negative numbers using a pre-specified number N as a biasing value. It is a nonweighted code. 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. (i.e. 0 − Excess Value)
• The greatest binary number represents the largest value. (i.e. 2 ^N+1 − Excess Value − 1)

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

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. In addition, when the sum of two XS-3 digits is greater than 9, the carry bit of a four bit adder will be set high. This works because, when adding two numbers that are greater or equal to zero, an "excess" value of six results in the sum. Since a four bit integer can only hold values 0 to 15, an excess of six means that any sum over nine will overflow.

Adding Excess-3 works on a different algorithm than non-biased decimal coding or regular binary positional system numbers. When you add two XS-3 numbers together, the result is not an XS-3 number. For instance, when you add 1 and 0 in XS-3 the answer seems to be 4 instead of 1. In order to correct this problem, when you are finished adding each digit, you have to remove the extra bias by subtracting binary 0011 (decimal 3 in unbiased binary) if the resulting digit is less than decimal 10 and subtracting binary 1101 (decimal 13 in unbiased binary), if an overflow has occurred. Note that, in 4-bit binary, subtracting binary 1101 is equivalent to adding 0011 and vice-versa.

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 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 dataflow;

See also

• Gray code
• Offset binary