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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?

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3 Answers

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It really depends on the CPU.

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

More info can be found here:

and here:

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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.

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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 .

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