Accurate Horner methods in real and complex floating-point arithmetic

In this article, we derive accurate Horner methods in real and complex floating-point arithmetic. In particular, we show that these methods are as accurate as if computed in k-fold precision and then rounded into the working precision. When k is two, our methods are comparable or faster than the existing compensated Horner routines. When compared to multi-precision software, such as MPFR and MPC, our methods are significantly faster, up to k equal to eight, that is, up to 489 bits in the significand.