问题
What order of magnitude difference should I be expecting for a subtraction between two theoretically equal double precision numbers?
I have two double precision arrays. They are expected to be theoretically same. They are both calculated by two completely different methodologies, so there is some numerical difference between them. I checked them element by element and my maximum difference is coming out to be 6.5557799910909154E-008. My boss says that for a double precision this is a very high difference, but I thought that if the difference if of the order of E-008, then its alright.
Thank you, Pradeep
回答1:
Double precision floating point has the following format
- Sign bit: 1 bit
- Exponent width: 11 bits
- Significand precision: 53 bits (52 explicitly stored)
This gives from 15 - 17 significant decimal digits precision. If a decimal string with at most 15 significant decimal is converted to IEEE 754 double precision and then converted back to the same number of significant decimal, then the final string should match the original; and if an IEEE 754 double precision is converted to a decimal string with at least 17 significant decimal and then converted back to double, then the final number must match the original.
Single precision floating point has the following format
- Sign bit: 1 bit
- Exponent width: 8 bits
- Significand precision: 24 (23 explicitly stored)
This gives from 6 to 9 significant decimal digits precision (if a decimal string with at most 6 significant decimal is converted to IEEE 754 single precision and then converted back to the same number of significant decimal, then the final string should match the original; and if an IEEE 754 single precision is converted to a decimal string with at least 9 significant decimal and then converted back to single, then the final number must match the original.
The maximum difference you are encountering indicates a loss of precision akin to converting to single precision.
Do you know which of the two methods is more accurate? Is it a trade-off between speed of computation and precision that is the main difference or is one of the algorithms less numerically stable? What is the precision of the inputs? A difference of 8 decimal digits of precision may not be relevant if your inputs aren't that precise... or it could mean missing Mars on a planetary trajectory.
来源：https://stackoverflow.com/questions/15493263/order-of-magnitude-for-double-precision