Hi Fredrico, See the discussion @
https://stackoverflow.com/questions/52736011/instruction-fyl2xp1 here is a relevant excerpt: The Taylor series for log(x) is usually done about x = 1. So every term will have x - 1. If you're trying to compute log(x + 1) for a very small x, a direct call as log(x + 1) will result in x + 1 - 1 which will drop all the low-order bits of x - thereby losing precision if x is really small. A built-in log(x+1)can then elide this x + 1 - 1 step and preserve the full precision > Le 3 mai 2020 à 08:52, Federico Miyara <fmiy...@fceia.unr.edu.ar> a écrit : > > > Dear all, > > I was comparing the accuracy of FFT and two exact formulas for the FFT of a > complex exponential and I was first surprised by a relative accuracy of only > 10^-13 for N = 4096, but on second thought it may be related to arithmetic > errors due to about N*log2(N) sums and products. > > But I was much more surprised to detect similar errors between different > exact formulas. These formulas involve a few instances of exponentials so I > conjectured that the problem may be related to the exponential accuracy. When > trying to find some information about accuracy in the documentation I found > none. > > The only mention in the elementary function set to accuracy appears in > log1p(), a strange function equal to log(1+x), which is seemingly included to > fix some accuracy problem of the natural logarithm very close to 1. Intuition > suggests that near 1 the Taylor approximation for log(1+x) should work very > well. I guess that is what log1p() does, so I wonder why a function such as > log1p is really necessary. It seems more reasonable to internally detect the > favorable situation and switch the algorithm to get the maximum attainable > accuracy. So if one needs an accurate log(1+x) function, one would just type > log(1+x)! > > But regardless of this discusion, I think it would very useful some hints > about accuracy in the help pages of elementary and other functions. > > For instance, with format(25) > > --> exp(10) > ans = > 22026.465794806717894971 > > while the Windows calculator (which is generally accurate to the last shown > digit) yields > > 22026.465794806716516957900645284 > > The underlined digits are the least significant ones common to both > solutions. Scilab shows up to 25 digits, but only the first 16 of them are > accurate. > > Regards, > > Federico Miyara > > _______________________________________________ > users mailing list > users@lists.scilab.org > https://antispam.utc.fr/proxy/1/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/lists.scilab.org/mailman/listinfo/users
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