| Re: Inaccuracies [message #5307 is a reply to message #5303] |
Tue, 14 November 1995 00:00   |
wclodius
Messages: 3
Registered: November 1995
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Junior Member |
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Some points:
1. It is impossible in finite binary arithmetic to represent any fraction
that is not an integer multiple of a power of two. The fpu implementation
limits the integer multiples and powers of twos available for such
representations to a finite set whose size depends on whether single or
double precision is used.
2. Division by 10 involves division by a non-power of two, e.g., the
prime number five. Therefore, numbers given to the first decimal place,
e.g., 0.1, cannot in general be represented exactly, multiples of 1.0 and
0.5 are exceptions. The representation of most of the numbers therefore
will be in error by a fraction of a bit. The sum of numbers given to the
first decimal place therefore, cannot, in general, be done exactly in
either single or double precision.
3. It is possible that depending. either on the average effects of
rounding and on the order of calculations, e.g., whether the sum goes from
smallest to largest, or vice versa, the result might accidently be the
"correct" answer. On average on such a simple problem, I suspect that
single and double precision will loose the same number of bits, but
because the double precision has more initially significant bits the
relative error in double precision will usually be much smaller than in
single precision.
4. It is not generally possible for a compiler or interpreter to know the
order of operations which minimizes the error, that requires a knowlege
about the detailed properties of the specific set of numbers subject to
the operations. It is sometimes possible to use your specific knowlege of
the numbers' properties to force the compiler to perform operations in a
specific order that minimizes errors, e.g., add the numbers in the order
smallest magnitude to largest magnitude to minimize roundoffs, or if the
number come in pairs with opposite signs, add the pairs together first.
Both methods have runtime performance hits.
5. The conversion in almost any implementation of C, Fortran, or IDL,
from the base two internal representation to the base ten output,
typically involves rounding, optimally to within the greater of the least
significant bit of the representation or the least significant digit of
the output, although that is quality of implementation dependent. This
rounding is an additional source of error that might or might not result
in the "correct" answer. For free format the least significant bit is
comparable in magnitude to the least significant digit in a high quality
implementation. For the example proplem, however, this is unlikely to be
a significant source of error in the output, as the higher order bits are
subtractted out before the output is generated.
--
William B. Clodius Phone (505) 665-9370
Los Alamos Natl. Lab. NIS-1 FAX (505) 665-7395
PO Box 1663, MS-D466 Group Office (505) 667-2701
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