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Re: Mathematica 20x slower than Java at arithmetic/special functions, is
Relevant to this thread -- maybe? (I'm not qualified to say) Some years ago I was doing some optical beam modal expansions involving numerical expansions of sums over Hermite-gaussian, Laguerre-gaussian, and Legendre functions with indexes running from n=0 up to, in extreme cases, n=60 or 70 -- just doing them as 'brute force' Mathematica Sums: no compiling (and no checking on timing). In some cases I wanted to observe effects of small variations of function arguments about special values like z = Pi, but without getting diverted into any attempts at doing first order analytical perturbations or expansions of the sums or their individual terms. Soon became apparent that if I evaluated, say, a 64-term sum for arguments z = 0.998 Pi, 0.999 Pi, Pi, 1.001 Pi, 1.002 Pi, I could see obvious numerical round-off errors in the results. But if I did the exact same calculations except writing the arguments as z = 998 Pi/1000, 999 Pi/1000, Pi, 1001 Pi/1000, 1002 Pi/ 1000, I got results that appeared to be accurate to the final digit in the results. I assumed this meant that Mathematica detected these exact inputs, and automatically switched to a (slower) exact arithmetical evaluation of the terms in the series (??). In the cases at issue here, maybe Mathematic does something similar -- and Java doesn't?