3.4 Binary-Coded Decimals (BCD)

52 Chapter 3

Figure 3-9 Packed and Unpacked BCD

3.4.1 Floating-Point BCD

Unlike the floating-point binary numbers, binary-coded decimal representations and

BCD arithmetic have not been explicitly described in a formal standard. Each machine or software package stores and manipulates BCD numbers in a unique and often incompatible way. Some machines include packed decimal formats, which are sign-magnitude BCD representations. These integer formats are useful for conversions and input-output operations. For performing arithmetic calculations a float- ing-point BCD encoding is required. This approach provides all the advantages of floating-point as well as the accuracy of decimal encodings.

The BCD floating-point format which we call BCD12 is shown Figure 3-8.

sign of number (1 BCD digit) sign of exponent (1 BCD digit)

exponent (4 BCD digits) significand (18 BCD digits)

S s e e e e m m m m m m m m m m m m m m m m m m

Figure 3-10 Map of the BCD12 Format

BCD12 requires 12 bytes of storage and is described as follows:

1.The sign of the number (S) is encoded in the left-most packed BCD digit. Therefore, the first four bits are either 0000B (positive number) or 0001B (negative number).

2.The sign of the exponent is represented in the four low-order bits of the first byte. The sign of the exponent is also encoded in one packed BCD digit. As is the case with the sign of the number field, the sign of the exponent is either 0000B (positive exponent) or 0001B (negative exponent)

3.The following two bytes encode the exponent in four packed BCD digits. The decimal range of the exponent is 0000 to 9999.

4.The remaining nine bytes are devoted to the significand field, consisting of 18 packed BCD digits. Positive and negative numbers are represented with a significand normal-

ized to the range 1.00...00 to 9.00...99. The decimal point following the first significand digit is implied. The special value 0 has an all-zero significand.

5.The special value FF hexadecimal in the number’s sign byte indicates an invalid number.

The structure of the BCD12 format is described in Table 3.4.

Table 3.4

Field Structure of the BCD12 Format

Notes:

1.The significand is scaled (normalized) to a number in the range 1.00..00 to 9.99..99.

2.The encoding for the value zero (0.00..00) is a special case.

3.The special value FFH in the sign byte indicates an invalid number.

The BCD12 format, as is the case in all BCD encodings, does not make ideal use of the available storage space. In the first place, each packed BCD digit requires four bits, which in binary could serve to encode six additional combinations. At a byte level the wasted space is of 100 encodings (BCD 0 to 99) out of a possible 256 (0 to FFH). The sign field in the BCD12 format is wasteful since only one binary digit is actually required for storing the sign. Regarding efficient use of storage, BCD formats cannot compete with floating-point binary encodings. The advantages of BCD representations are a greater ease of conversion into decimal forms, and the possibility of using the processors’ BCD arithmetic instructions.