The complementary error function, erfc, is a special functions closely related to the standard normal distribution. It is frequently used in statistics and the natural sciences (e.g. diffusion problems) where the "tails" of this distribution need to be considered, and use of the error function, erf, is therefore not suitable.
The complementary error function was made available in the ISO C99 standard math library as the functions erfcf, erfc, and erfcl; these were subsequently adopted into ISO C++ as well. Thus source code can readily be found in open-source implementations of that library, for example in glibc.
However, many existing implementations are scalar in nature, while modern processor hardware is SIMD-oriented (either explicitly, as in x86 CPU, or implicitly, as in GPUs). For performance reasons, a vectorizable implementation is therefore highly desirable. This means branches need to be avoided, except as part of select assignment. Likewise, extensive use of tables is not indicated, as parallelized lookup is often inefficient.
How would one go about constructing an efficient vectorizable implementation of the single-precision function erfcf()? The accuracy, as measured in ulp, should be roughly the same as glibc's scalar implementation, which has a maximum error of 3.12575 ulps (determined by exhaustive testing). The availability of fused multiply-add (FMA) can be assumed, as all major processor architectures (CPUs and GPUs) offer it at this time. While handling of floating-point status flags and errno can be ignored, denormals, infinities, and NaNs should be handled in accordance with the IEEE 754 bindings for ISO C.