[Scilab-users] log and log1p

[Federico Miyara]

Stéphane,

Thanks. I think I get the idea: log(1+x) allows values much closer to 1 
than possible with log(x) because x could fall down way below %eps, 
since the full folating point range for small values would be available 
(about 1e-323).

But then the same would be true for any function that happens to be 0 at 
some point, such as cos(x) at %pi/2, or besselj(x) at any of its zeros.

Besides, for any value for which log1p() is relevant (in the sense that 
it would differ from an optimized log(x) that is accurate up to 1 + %eps 
or so) probably log1p(x) wouldn't differ from x, since the next digit, 
corresponding to the x^2 Taylor term cannot fit in the mantissa of a 
double different from x.

For instance (as per Wolfram Alpha site --truncated),

log(1 + 1e-16) = 
9.99999999999999950000000000000003333333333333333083333333333333353333333333333331... 
× 10^-17

Attempting to represent this with double, it yields 0.0000000000000001

No difference from 1e-16 since the next different digit would produce a 
difference below %eps.

log1p() would make sense in case the result could be given with extended 
precision

Regards,

Federico Miyara



On 03/05/2020 05:41, Stéphane Mottelet wrote:
> 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 informationabout 
>> accuracyin 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 wouldjusttype 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  =
>>    22026.4657948067_1_7894971
>>
>> while the Windows calculator (which is generally accurate to the last 
>> shown digit) yields
>>
>> 22026.4657948067_1_6516957900645284
>>
>> 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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