Efficiently computing (a - K) / (a + K) with improved accuracy

Question

In various contexts, for example for the argument reduction for mathematical functions, one needs to compute (a - K) / (a + K), where a is a positive v

Answer 1

I don't really have an answer (proper floating point error analyses are very tedious) but a few observations:

• Fast reciprocal instructions (such as RCPSS) are not as accurate as division, so you may see a reduction in accuracy if using these.
• m is computed exactly if a ∈ [0.5×K_b, 2¹⁺ⁿ×K_b), where K_b is the power of 2 below K (or K itself if K is a power of 2), and n is the number of trailing zeros in the significand of K (i.e. if K is a power of 2, then n=23).
• This is similar to a simplified form of the div2 algorithm from Dekker (1971): to expand the range (particularly the lower bound), you'll probably have to incorporate more correction terms from this (i.e. store m as the sum of 2 floats, or use a double).