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Excel maintains 15 figures in its numbers, but they are not always accurate; mathematically, the bottom line should be the same as the top line, in 'fp-math' the step '1 + 1/9000' leads to a rounding up as the first bit of the 14 bit tail '10111000110010' of the mantissa falling off the table when adding 1 is a '1', this up-rounding is not undone when subtracting the 1 again, since there is no ...
Variable length arithmetic represents numbers as a string of digits of a variable's length limited only by the memory available. Variable-length arithmetic operations are considerably slower than fixed-length format floating-point instructions.
Various software patches were produced by manufacturers to work around the bug. One specific algorithm, outlined in a paper in IEEE Computational Science & Engineering, is to check for divisors that can trigger the access to the programmable logic array cells that erroneously contain zero, and if found, multiply both numerator and denominator by 15/16.
Because Excel is widely used, it has been attacked by hackers. While Excel is not directly exposed to the Internet, if an attacker can get a victim to open a file in Excel, and there is an appropriate security bug in Excel, then the attacker can gain control of the victim's computer. [125] UK's GCHQ has a tool named TORNADO ALLEY with this purpose.
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These n bits are the remainder of the division step, and will also be the value of the CRC function (unless the chosen CRC specification calls for some postprocessing). The validity of a received message can easily be verified by performing the above calculation again, this time with the check value added instead of zeroes.
The original theoretical decoder generated potential polynomials based on subsets of k (unencoded message length) out of n (encoded message length) values of a received message, choosing the most popular polynomial as the correct one, which was impractical for all but the simplest of cases.