Finding the "truncated" polynomial that is closest to a function
<|reference_start|>Finding the "truncated" polynomial that is closest to a function: When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite - and sometimes small - number of bits: this is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree-$i$ coefficient has at most $m_i$ fractional bits (in other words, it is a rational number with denominator $2^{m_i}$). We provide a general method for finding the best polynomial approximation under this constraint. Then, we suggest refinements than can be used to accelerate our method.<|reference_end|>
@article{brisebarre2003finding,
    title={Finding the "truncated" polynomial that is closest to a function},
    author={Nicolas Brisebarre and Jean-Michel Muller},
    journal={arXiv preprint arXiv:cs/0307009},
    year={2003},
    archivePrefix={arXiv},
    eprint={cs/0307009},
    primaryClass={cs.MS}
}