[Scilab-users] log and log1p

[Stéphane Mottelet]

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 <fmiyara at 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
> 
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