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Red Hat Engineer Improves Math Performance of Glibc
jones_supa writes: Siddhesh Poyarekar from Red Hat has taken a professional look into mathematical functions found in Glibc (the GNU C library). He has been able to provide an 8-times performance improvement to slowest path of pow() function. Other transcendentals got similar improvements since the fixes were mostly in the generic multiple precision code. These improvements already went into glibc-2.18 upstream. Siddhesh believes that a lot of the low hanging fruit has now been picked, but that this is definitely not the end of the road for improvements in the multiple precision performance. There are other more complicated improvements, like the limitation of worst case precision for exp() and log() functions, based on the results of the paper Worst Cases for Correct Rounding of the Elementary Functions in Double Precision (PDF). One needs to prove that those results apply to the Glibc multiple precision bits.
What is perhaps a bit of irony of history, even for humans a lookup table is faster and more precise than manually calculating it via formula. That is why they published books of logarithms. Using interpolation you can even stretch out the precision to several more digits. With a table of values in memory you can also narrow down the inputs to Newton's method and calculate any differentiable function very quickly to an arbitrary precision. With some functions the linear approximation is so close that you can reduce it in just a few cycles.
Even in most trigonometric functions there is a simple table upon which the angle addition formulas are used to get the other values[an old example].
Given the size of most operating systems, where 8k of ram is hardly noticed (most gifs are larger than this), I am actually quite surprised that the lookup table method is not more used. It would seem one of the first things to put in cache on your ALU.