Finding the "truncated" polynomial that is closest to a function

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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 mᵢ fractional bits (in other words, it is a rational number with denominator 2ᵐⁱ). 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.