, 
is a transcendental number for any finite floating-point value x. This means there could be some floating-point number whose logarithm is 
, where the braces indicate bits beyond the 53 bits of double precision, and where the sequence of 1-bits in the braces is thousands or millions of bits long. That is, the logarithm of a floating-point value could be arbitrarily close to a half-way case. An algorithm that computed successively more bits, but always with some uncertainty in the last bit computed, would have to compute beyond the last 1-bit in the long sequence in order to ascertain that the result was correctly rounded.
Because it is possible that some value could require computation beyond the time a user is willing to wait or beyond the memory capacity of a particular computer, and because the double numbers are too numerous to check exhaustively, it is impossible to prove beyond mathematical doubt that an implementation of a transcendental function for double or long double arguments is correctly rounded.
The fate of rounded values excruciatingly close to half-way case is left in the balance. The best algorithms are known to be correctly rounded for all but one in a trillion cases; they are not known to fail for the others.
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