Efficient system of binary encoding for decimal digits used in decimal floating point
This article is about an efficient method for BCD encoding. For a simpler, but less efficient method, see Packed decimal.
Densely packed decimal (DPD) is an efficient method for binary encoding decimal digits.
The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in significant wastage of binary data bandwidth (since four bits can store 16 states and are being used to store only 10), even when using packed BCD. Densely packed decimal is a more efficient code that packs three digits into ten bits using a scheme that allows compression from, or expansion to, BCD with only two or three hardware gate delays.[1]
The densely packed decimal encoding is a refinement of Chen–Ho encoding; it gives the same compression and speed advantages, but the particular arrangement of bits used confers additional advantages:
Compression of one or two digits (into the optimal four or seven bits respectively) is achieved as a subset of the three-digit encoding. This means that arbitrary numbers of decimal digits (not only multiples of three digits) can be encoded efficiently. For example, 38 = 12 × 3 + 2 decimal digits can be encoded in 12 × 10 + 7 = 127 bits – that is, 12 sets of three decimal digits can be encoded using 12 sets of ten binary bits and the remaining two decimal digits can be encoded using a further seven binary bits.
The subset encoding mentioned above is simply the rightmost bits of the standard three-digit encoding; the encoded value can be widened simply by adding leading 0 bits.
All seven-bit BCD numbers (0 through 79) are encoded identically by DPD. This makes conversions of common small numbers trivial. (This must break down at 80, because that requires eight bits for BCD, but the above property requires that the DPD encoding must fit into seven bits.)
The low-order bit of each digit is copied unmodified. Thus, the non-trivial portion of the encoding can be considered a conversion from three base-5 digits to seven binary bits. Further, digit-wise logical values (in which each digit is either 0 or 1) can be manipulated directly without any encoding or decoding being necessary.
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