No, not by finding a polynomial-time algorithm for SAT. Just by using brute-force search.
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I'm making the same assumptions about the ordering that people always make when they say SAT is NP-complete. Computing the nth bit of a number is not the same as computing a rational approximation to that number.
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Knowing the nth bit of a real number is typically useless if you do not also know the first n-1 bits. The algorithm I had in mind was to compute the bits 1 at a time by brute-force search until you find enough bits to come within the desired precision.
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