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Instead of using Newton's method to compute numeric square roots, use the Karatsuba square root algorithm, which performs better for numbers of all sizes. In practice, this is 3-5 times faster for inputs with just a few digits and up to around 10 times faster for larger inputs. Also, the new algorithm guarantees that the final digit of the result is correctly rounded, since it computes an integer square root with truncation, containing at least 1 extra decimal digit before rounding. The former algorithm would occasionally round the wrong way because it rounded both the intermediate and final results. In addition, arrange for sqrt_var() to explicitly support negative rscale values (rounding before the decimal point). This allows the argument reduction phase of ln_var() to be optimised for large inputs, since it only needs to compute square roots with a few more digits than the final ln() result, rather than computing all the digits before the decimal point. For very large inputs, this can be many thousands of times faster. In passing, optimise div_var_fast() in a couple of places where it was doing unnecessary work. Patch be me, reviewed by Tom Lane and Tels. Discussion: https://postgr.es/m/CAEZATCV1A7+jD3P30Zu31KjaxeSEyOn3v9d6tYegpxcq3cQu-g@mail.gmail.com |
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