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    \$\begingroup\$ This is equal (if I've understood it right) to 1/φ (reciprocal of golden ratio). If you want us to actually use the Fibonacci numbers in the calculation, you should specify. If not, there's certainly languages where φ is a builtin. (not APL for a change) \$\endgroup\$ Commented Jan 9, 2014 at 22:40
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    \$\begingroup\$ @marinus Changed. \$\endgroup\$ Commented Jan 9, 2014 at 22:55
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    \$\begingroup\$ @marinus, it's not equal to 1/phi. It does have a closed form, but it's quite tricky. Mathematica probably has a built-in, but I doubt many other languages do. \$\endgroup\$ Commented Jan 9, 2014 at 23:22
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    \$\begingroup\$ @OP, "highest accuracy possible" is a useless winning criterion. Anyone who has a language which supports arbitrary decimal precision, or who can be bothered to write the support for it, can make an implementation with a precision parameter and then engage in an edit war to increase that parameter. It would make more sense to ask for a function which takes the precision as a parameter. Judging on speed is also tricky, because it depends on many factors (CPU model, RAM available, system load, ...). \$\endgroup\$ Commented Jan 9, 2014 at 23:28
  • \$\begingroup\$ @PeterTaylor Is this better? \$\endgroup\$ Commented Jan 9, 2014 at 23:41