We propose an alternative approach to the construction of fitting functions to the nonlinear matter power spectrum extracted from $N$-body simulations based on the relative matter power spectrum $delta(k,a)$, defined as the fractional deviation in the absolute matter power spectrum produced by a target cosmology away from a reference $Lambda$CDM prediction. From the computational perspective, $delta(k,a)$ is fairly insensitive to the specifics of the simulation settings, and numerical convergence at the 1%-level can be readily achieved without the need for huge computing capacity. Furthermore, $delta(k,a)$ exhibits several interesting properties that enable a piece-wise construction of the full fitting function, whereby component fitting functions are sought for single-parameter variations and then multiplied together to form the final product. Then, to obtain 1%-accurate absolute power spectrum predictions for any target cosmology only requires that the community as a whole invests in producing one single ultra-precise reference $Lambda$CDM absolute power spectrum, to be combined with the fitting function to produce the desired result. To illustrate the power of this approach, we have constructed the fitting function RelFit using only five relatively inexpensive $w$CDM simulations (box length $L=256 h^{-1}$Mpc, $N=1024^3$ particles, initialised at $z_i=49$). In a 6-parameter space spanning ${omega_m,A_s,n_s,w,omega_b,h}$, the output relative power spectra of RelFit are consistent with the predictions of the CosmicEmu emulator to 1% or better for a wide range of cosmologies up to $ksimeq 10$/Mpc. Thus, our approach could provide an inexpensive and democratically accessible route to fulfilling the 1%-level accuracy demands of the upcoming generation of large-scale structure probes, especially in the exploration of non-standard or exotic cosmologies on nonlinear scales.
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