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Suppose I have a 32-bit integer $x$, I want to find $\{ x_i \}_{i \in 1\dots\ell}$ such that $x = e + \sum_{i=1}^\ell x_i \cdot 2^{32 - B\cdot i}$ where the error $e$ is as small as possible. The parameter $\ell$ is the level of the decomposition and $B$ is the base of the decomposition. Increasing $\ell$ and decreasing $B$ will result in a more accurate decomposition.

Is there an algorithm that solves this kind of problem?

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  • $\begingroup$ Is $B$ an integer? $\endgroup$
    – xskxzr
    Commented Mar 23, 2022 at 2:32
  • $\begingroup$ Some requirements must be missing. Can't you set $x_\ell = x/2^{32-B\ell}$ and all other $x_i=0$? Do you require the $x_i$'s to be integers? Why is this anything other than a base conversion problem? $\endgroup$
    – D.W.
    Commented Mar 23, 2022 at 6:25
  • $\begingroup$ What are $x_i$s? I assume those are integers, right? If so - then using only one integer, when $i$ is the largest - will result in the same accuracy (since all other values are integer multiples of it), so you can solve this algebraically. So maybe restricting $x_i$ to some range (or even bits) would be much more interesting $\endgroup$
    – nir shahar
    Commented Mar 23, 2022 at 6:26
  • $\begingroup$ Assuming that L is fixed, and the B_i are fixed, you would indeed just pick I so that B_i is largest B_i, let x_i = x / 2^(32 - B_i), rounded to the nearest integer, and all other x_k are zero. Error is at most half of 2^(32 - B_i). $\endgroup$
    – gnasher729
    Commented Aug 24, 2022 at 13:48

1 Answer 1

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Assuming that $B$ and the $x_i$ are integers, you can recast the question as

$$\frac x{2^{32}}\approx\sum_{i=1}^l x_iP^i$$ where $P:=2^{-B}$, and you look for the best approximation of $\dfrac x{2^{32}}<1$ in base $2^B$ using $l$ digits.

Just write the binary representation of that number and get the digits in $l$ slices of $B$ bits.

E.g. $\frac{123456789_d}{2^{32}}=0.00000111010110111100110100010101_b=0.01655715052_o$ ($o$ for octal) gives all solutions for $B=3$.

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