Synthesizing Efficient Low-Precision Kernels

Dandelion: Certified Approximations of Elementary Functions

“…Chip sizes and energy budgets in these devices are usually small, and thus using a fully-fledged floating-point unit is not always possible. As an alternative, code is often implemented in fixed-point arithmetic, which, however, does not come with standard and efficient library implementations of elementary functions [21]. Hence, an engineer may approximate the function cos on line 8 in Figure 1a with a custom polynomial approximation…”

“…Dandelion: Certified Approximations of Elementary Functions shown in Figure 1b, for instance using the state-of-the-art synthesis tool Daisy [21] 3 .…”

“…As a Remez-like algorithm, we use the fpminimax [9] function in Sollya. Our pipeline is benchmarked on numerical kernels taken from the FPBench benchmark suite [13], and the benchmarks used by an unverified extension of Daisy with approximations for elementary functions [21]. These benchmarks represent kernels as they occur in e.g.…”

“…Dandelion can be used as a verifier for any Remez-like algorithm. In our evaluation, we use Dandelion to certify a number of approximations generated from FPBench [13] and the work by Izycheva et al [21]. Our evaluation shows that certificate checking in Dandelion is fast, and that Dandelion certifies, for an elementary function f and polynomial p, approximation errors on the same order of magnitude as the infinity norm (max x∈I(x) |f (x) − p(x)|).…”

“…However, these can be inefficient for applications that do not need quite as much accuracy, or that only need to work for a limited set of inputs [15,22]. Furthermore, many applications, especially in the embedded systems domain, operate with fixed-point arithmetic, for which efficient library approximations do not exist [21].…”