What algorithm is used by computers to calculate logarithms?

Question

I would like to know how logarithms are calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that logarithms are calculated directly from the hardware.

So the question is: what algorithm is used by computers to calculate logarithms?

Answer 1

It really depends on the CPU.

For intel IA64, apparently they use Taylor series combined with a table.

More info can be found here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5177

and here: http://www.computer.org/csdl/proceedings/arith/1999/0116/00/01160004.pdf

Answer 2

Read the docs and the source of the cephes library for instance. Try also these books:

• J. F. Hart et al., Computer Approximations, Wiley, 1968.
• W. J. Cody, Jr. and W. Waite. Software Manual for the Elementary Functions. Prentice Hall, 1980.
• J.-M. Muller, Elementary Functions, Algorithms and Implementation, SIAM, 2006.

See also https://stackoverflow.com/questions/2169641/where-to-find-algorithms-for-standard-math-functions.

Answer 3

All methods I have seen reduce first by dividing by the power of $2$ in the exponent and adding that exponent times $\ln(2)$ to the result. From there, the two most common methods I have seen are Taylor series for $\ln(1+x)$ and a variant of the CORDIC algorithm.

J. M. also brings up Padé approximants which I have seen used in some calculator implementations.