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In this Wolfram Conference talk from last year, "Faster Arbitrary Precision Computation of Elementary Functions", they talk about R&D for algorithms to numerically evaluate elementary functions using arbitrary precision arithmetic. Has this technology been incorporated into Mathematica? If so, when? If not, when?

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    $\begingroup$ Yes, as of 11.0.0. $\endgroup$ – ilian Aug 20 '16 at 23:08
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    $\begingroup$ @ilian great! Is there documentation where I can read more about this? $\endgroup$ – QuantumDot Aug 21 '16 at 4:58
  • $\begingroup$ @ilian I ask for more info because the documentation pages near the bottom of Sqrt, Log, Exp, etc. do not mention changes in v11. $\endgroup$ – QuantumDot Aug 21 '16 at 13:54
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    $\begingroup$ Not aware of any documentation changes, but keep in mind that the functions still do the same job, they have just been reimplemented for efficiency. Such internal optimizations are rarely documented and I can't imagine any update being as informative as the presentation linked. $\endgroup$ – ilian Aug 21 '16 at 14:37
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    $\begingroup$ @ilian Thanks. My principal interests in Mathematica are in enhancements of core language and function optimization. That such an important update wasn't even mentioned in the new features list is a shame. I do feel that if such things were better highlighted, a lot of people (especially the loyal/experienced ones) would be more excited for new versions. $\endgroup$ – QuantumDot Aug 21 '16 at 17:49
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Posting my earlier comment as an answer:

Yes, as of 11.0.0.

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  • $\begingroup$ I am happy to learn that this kind of development is still taking place! $\endgroup$ – Mr.Wizard Aug 22 '16 at 23:04
  • $\begingroup$ Is there any benchmark of this? I tried the below code in v10.3 and v11.0.0 but did not see any significant change: AbsoluteTiming[N[ArcTan[123456], 10^5];] and AbsoluteTiming[N[Log[123456], 10^5];] (on different computer so not so precise comparison. Nevertheless if the change is big it should show up). $\endgroup$ – Yi Wang Aug 23 '16 at 8:08
  • $\begingroup$ @YiWang, In my mac I tried N[Cos[234], 10^6]; // AbsoluteTiming and N[Sin[134], 10^4]; // AbsoluteTiming, on versions 10.3 and 11.0, and I do find an improvement: 12.01 s vs 10.98 s for Cos and and 0.202 s vs 0.011 s for Sin. $\endgroup$ – QuantumDot Aug 24 '16 at 8:37
  • $\begingroup$ @YiWang Yes, I would again refer to the presentation for example results. The exact speedup will depend on the function called, the argument range and the precision requested. Furthermore, there is a cost associated with going through the main evaluator which can obscure the performance gained in certain examples with shorter runtimes. Accurate benchmark testing therefore required some C code instrumentation in the kernel. $\endgroup$ – ilian Aug 24 '16 at 21:15

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