Theorems on Efficient Argument Reductions

A commonly used argument reduction technique in el-ementary function computations begins with two positive floating point numbers α and γ that approximate (usually irrational but not necessarily) numbers 1/C and C, e.g., C = 2π for trigonometric functions and ln 2 for ex. Given an argument to the function of interest it extracts z as de-fined by xα = z + ς with z = k2−N and |ς | ≤ 2−N−1, where k,N are integers and N ≥ 0 is preselected, and then computes u = x − zγ. Usually zγ takes more bits than the working precision provides for storing its significand, and thus exact x − zγ may not be represented exactly by a float-ing point number of the same precision. This will cause per-formance penalty when the working precision is the highest available on the underlying hardware and thus consider-able extra work is needed to get all the bits of x − zγ right. This paper presents theorems that show under mild condi-tions that can be easily met on today’s computer hardware and still allow α ≈ 1/C and γ ≈ C to almost the full work-ing precision, x − zγ is a floating point number of the same precision. An algorithmic procedure based on the theorems is obtained. The results will enhance performance, in par-ticular on machines that has hardware support for fused-multiply-add (fma) instruction(s).