Math precision requirements of C and C++ standard

Question

Do the C and C++ standards require the math operations in math.h on floating points (i.e. sqrt, exp, log, sin, ...) to return numerically best solution?

For a given (exact and valid) input there can obviously in general not be an exact floating point output from these functions. But is the output required to be the representable value nearest to the mathematically exact one?

If not, are there any requirements on precision whatsoever (possibly platform-specific / in other standards ?), so that I am able to make worst-case estimates of calculation errors in my code? What are typical limits on numerical errors of modern implementations?

Answer 1

No, and for good reason. In general, you'd need an infinite precision (and infinite time) to determine the exact mathematical result. Now most of the times you need only a few extra iterations to determine sufficient bits for rounding, but this number of extra bits depend on the exact result (simply put: when the result is close to .5 ULP). Even determining the extra number of iterations required is highly non-trivial. As a result, requiring exact results is far, far slower than a pragmatic approach.