  The Turing completeness result for continuous chemical reaction networks
(CRN) shows that any computable function over the real numbers can be computed
by a CRN over a finite set of formal molecular species using at most
bimolecular reactions with mass action law kinetics. The proof uses a previous
result of Turing completeness for functions defined by polynomial ordinary
differential equations (PODE), the dualrail encoding of real variables by the
difference of concentration between two molecular species, and a back-end
quadratization transformation to restrict to elementary reactions with at most
two reactants. In this paper, we present a polynomialization algorithm of
quadratic time complexity to transform a system of elementary differential
equations in PODE. This algorithm is used as a front-end transformation to
compile any elementary mathematical function, either of time or of some input
species, into a finite CRN. We illustrate the performance of our compiler on a
benchmark of elementary functions relevant to CRN design problems in synthetic
biology specified by mathematical functions. In particular, the abstract CRN
obtained by compilation of the Hill function of order 5 is compared to the
natural CRN structure of MAPK signalling networks.
