The draft IEEE 754r standard proposed two encoding formats for decimal floating point numbers, the formats are referred to in the standard as binary encoding and decimal encoding.[1]. The binary encoding format is referred to by Intel[2] and others as Binary Integer Decimal (BID).
Format
A BID number has three fields, the sign the exponent and the significand.
If the top bits of the exponent are less than or equal binary 1110:
s = +1 (positive numbers) when the sign bit is 0
s = −1 (negative numbers) when the sign bit is 1
e = exponent − bias
m = significand.
The number has the value:
v = s × 10e × m
The exponent and significand are stored as binary integers.
If the top five bits of the exponent field has the encoding 11110 binary, the number represents + or - infinity, the sign bit determines which. Nan is represented by the tops five bits of the exponent field being set to 11111.
Cohort
A decimal floating point number can be encoded in several ways, the different ways represent different accuracies, for example 100.0 could be encoded as 1000*10-1, 100.00 could be encoded 10000*10-2. The set of possible encodings is called a cohort. If the result of a calculation is inexact the largest amount of significant data is preserved by selecting the cohort member with the largest integer that can be stored in the significand along with the required exponent.
Range
The proposed IEEEE754r standard limits the range of numbers to the much smaller range that can be represented by the other encoding described by the standard, Densely Packed Decimal.
| item | 32 bit | 64 bit | 128 bit |
|---|---|---|---|
| Storage bits | 32 | 64 | 128 |
| Trailing Significand bits | 20 | 50 | 110 |
| Significand bits | 23 | 53 | 113 |
| Combination bit | 11 | 13 | 17 |
| Exponent bits | 8 | 10 | 14 |
| Bias | 101 | 398 | 6176 |
| Standard emax | 96 | 384 | 6144 |
| Standard emin | 95 | 383 | 6143 |
Comparing BID numbers
BID numbers are lexicographically ordered. If NaNs are excluded, the can be compared (>, <, or ==) as sign and magnitude integers.
See Also
References
- ^ "DRAFT Standard for Floating Point Arithmetic P754" (PDF). 2006-10-04. Retrieved 2007-07-01.
- ^ "IEEE 754R Decimal Floating-Point Arithmetic: Reliable and Efficient Implementation for Intel® Architecture Platforms". Intel. February 15, 2007. Retrieved 2007-07-01.