log2() function from C standard library is inaccurate

[Falk Tannhäuser]

I noticed that the log2() function in GNU Octave returns inaccurate results
for many arguments that are exactly representable integer powers of 2.
After discussion on the Octave mailing list https://savannah.gnu.org/bugs/?42583
it occurs that Octave doesn't have this problem on most other platforms
(Linux, OS X) and that it is due to the fact that Octave usually uses
log2() from the local C library.

The following C++ program exposes the error:
_________________________________________________________________
// Compilation: g++ -Wall -Wextra -O3 -s testlog2.cxx -o testlog2
#include <limits>
#include <iostream>

#if 0 /////////////////////////////////////////////////////////
inline double log2(double x)
{
   double result;
   asm ("fyl2x" : "=t" (result) : "0" (x), "u" (1.0) : "st(1)");
   return result;
}
#else /////////////////////////////////////////////////////////
#include <math.h>
#endif ////////////////////////////////////////////////////////

int main()
{
   int n = std::numeric_limits<double>::min_exponent
         - std::numeric_limits<double>::digits;
   std::cout << "n = " << n << " ... "
             << std::numeric_limits<double>::max_exponent - 1
             << '\n';
   int errcnt = 0;
   for(double x = std::numeric_limits<double>::denorm_min();
       x < std::numeric_limits<double>::infinity();
       x *= 2, ++n)
   {
     if(n != log2(x))
     {
       std::cout << n << '\n';
       ++errcnt;
     }
   }
   std::cout << "Found " << errcnt << " errors.\n";
   return 0;
}
_________________________________________________________________

With GCC 4.9.0 or 4.8.3 for x86_64-pc-cygwin, erroneous results
are found for 441 values. This seems to be due to the fact that
log2 is implemented in terms of log (as something like log(x)/log(2)).
OTOH, when using the assembler version of log2 (just changing '#if 0'
to '#if 1' in above code); accurate results are always returned.

Falk

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