D and math, can you isolate this ?

[Basile B. via Digitalmars-d-learn]

On Tuesday, 20 September 2016 at 16:22:19 UTC, H. S. Teoh wrote:
> On Tue, Sep 20, 2016 at 12:35:18PM +0000, Basile B. via 
> Digitalmars-d-learn wrote: [...]
>> The problem is here:
>> https://github.com/BBasile/iz/blob/master/import/iz/math.d#L849
>> - f(x,c) = 1.0 - pow(1.0 - pow(x, 2.0/c), c * 0.5);
>> - c(f0.5)) = ?
>> 
>> Which means that I ask you if you can isolate c for
>> 
>> y = 1.0 - pow(1.0 - pow(0.5, 2.0/c), c * 0.5);
>> 
>> y is always f(0.5,c)
>
> I couldn't manage to solve it.  Nested exponentials are very 
> nasty to invert. :-(  At first, I thought it might be solvable 
> in terms of the Lambert W function (aka ProductLog) but I 
> couldn't manage to get the equation into the right form.  Then 
> I checked on Wolfram Alpha and it says "no result found in 
> terms of standard mathematical functions".
>
> That probably means the inverse cannot be expressed in terms of 
> elementary functions. Probably the only thing you can do is to 
> use some kind of numerical approximation, like some form of 
> Newton's method or some such, to find the value of c.
>
>
> T

Thanks for trying, you're not the first to tell me about the 
Newton's method...