Question

Are the old tricks (lookup table, approx functions) for creating faster implementations of sqrt() still useful, or is the default implementation as fast as it is going to get with modern compilers and hardware?

Answer 1

Rule 1: profile before optimizing

Before investing any effort in the belief that you can beat the optimizer, you must profile everything and discover where the bottleneck really lies. In general, it is unlikely that sqrt() itself is your bottleneck.

Rule 2: replace the algorithm before replacing a standard function

Even if sqrt() is the bottleneck, then it is still reasonably likely that there are algorithmic approaches (such as sorting distances by length squared which is easily computed without a call to any math function) that can eliminate the need to call sqrt() in the first place.

What the compiler does for you if you do nothing else

Many modern C compilers are willing to inline CRT functions at higher optimization levels, making the natural expression including calls to sqrt() as fast as it needs to be.

In particular, I checked MinGW gcc v3.4.5 and it replaced a call to sqrt() with inline code that shuffled the FPU state and at the core used the FSQRT instruction. Thanks to the way that the C standard interacts with IEEE 754 floating point, it did have to follow the FSQRT with some code to check for exceptional conditions and a call to the real sqrt() function from the runtime library so that floating point exceptions can be handled by the library as required by the standard.

With sqrt() inline and used in the context of a larger all double expression, the result is as efficient as possible given the constraints of of standards compliance and preservation of full precision.

For this (very common) combination of compiler and target platform and given no knowledge of the use case, this result is pretty good, and the code is clear and maintainable.

In practice, any tricks will make the code less clear, and likely less maintainable. After all, would you rather maintain (-b + sqrt(b*b - 4.*a*c)) / (2*a) or an opaque block of inline assembly and tables?

Also, in practice, you can generally count on the compiler and library authors to take good advantage of your platform’s capabilities, and usually to know more than you do about the subtleties of optimizations.

However, on rare occasions, it is possible to do better.

One such occasion is in calculations where you know how much precision you really need and also know that you aren’t depending on the the C standard’s floating point exception handling and can get along with what the hardware platform supplies instead.

Edit: I rearranged the text a bit to put emphasis on profiling and algorithms as suggested by Jonathan Leffler in comments. Thanks, Jonathan.

Edit2: Fixed precedence typo in the quadratic example spotted by kmm‘s sharp eyes.