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floating-point values so that all machines conforming to the standard produce the same
results for an operation. The operation must meet the following conditions:
1. Same input values,
2. Same rounding mode, and
3. Same precision.
The IEEE standard specifies not only the format of data items, but also defines:
1. The maximum allowable error that may be introduced during a calculation, and
2. The manner in which rounding of the result is performed.
However, the iEEE specification defines only the operation of some of the instructions
supported by the FPCP; those not specified by the IEEE standard are described in detail
in the following paragraphs. The following paragraphs discuss the accuracy of the calcu-
lations performed by the FPCP, grouping them as follows:
1. The IEEE specified operations and nontranscendental functions,
2. The transcendental functions, and
3. The IEEE specified conversions between binary and decimal real formats.
4.3.1 Arithmetic Instructions
The /EEE Specification for Binary Floating-Point Arithmetic specifies that the following
operations must be supported for each data format: add, subtract, multiply, divide, re-
mainder, square root, integer part, and compare. Conversions between the various data
formats are also required. In addition to these arithmetic functions, the FPCP also supports
the nontranscendental operations of: absolute value, get exponent, get mantissa, negate,
modulo remainder, scale exponent, and test. Since the IEEE specification defines the error
bounds to which all calculations are performed, the result obtained by any conforming
machine can be predicted exactly for a particular precision and rounding mode. The error
bound defined by the IEEE specification is one-half unit in the last place of the destination
data format in the round-to-nearest mode and one unit in the last place in the other rounding
modes.
The FPCP performs all calculations using a 67-bit mantissa for the intermediate results.
The three bits beyond the precision of the extended format allow the FPCP to perform all
calculations as if to infinite precision and then round the result to the desired precision
before storing it in the destination. By performing calculations in this manner, the final
result is always correct for the specified destination data format before rounding is per-
formed (unless an overflow or underflow error occurs). The specified rounding operation
then produces a number that is as close as possible to the infinitely precise intermediate
value and is still representable in the selected precision. An example of how the 67-bit
mantissa
allows the FPCP to meet the error bound of the JEEE speci~cation is as follows:
Mantissa 1 g r s
Intermediate Result: x.x ...... x00 1 0 0 (Tie Case)
Round-to-Nearest Result: x.x ...... x00
In this case, the least significant bit (1) of the rounded result is not incremented, even
though the guard bit (g) is set in the intermediate result. The IEEE standard specifies that
tie cases should be handled in this manner. Assuming that the destination data format is
FREESCALE
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MC68881/MC68882 USER'S MANUAL
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