Dear Naveen,

your problem depends on the degree of approximation you are searching. The HTF is quite simple to compute. Bby using a fit (from cubic to 10th degree) you can get various approximations of the function. The result depends of what you want to do with. For instance you have a good fit with the following 10th degree polyniomial function

y = 0.0001616 + 0.86635 * x - 0.00042018 * x^2 -

0.13333 * x^3 + 0.00016808 * x^4 + 0.012015 * x^5

- 2.286e-05 * x^6 - 0.00050388 * x^7 +

1.241e-06 * x^8 + 7.789e-06 * x^9 - 2.3167e-08 * x^10

But for you is it a "simple" approximation of the HTF? (its more simple to compute explicitely the THF itself... in my sense)

G.C.

Dear Naven, you can find a rational function to approximate tanh:

tanh(z)= Tanh[z] == z/(1 + z^2/(3 + z^2/(5 + z^2/(7 + z^2/(9 + z^2/(11+z^2/(13 )))))))

just look at  the web link:    

   http://functions.wolfram.com/ElementaryFunctions/Tanh/10/0001/ 

but may be this is more complicate than the standart polynomial  interpolation function described by Guy Caniaux from CNRM center.

regards

hubert

http://functions.wolfram.com/ElementaryFunctions/Tanh/10/0001/

Note that from Hubert's elegant solution, you have already a nice approximation with part of his rational function: x/(1.+x**2/(3.+x**2/(5.+x**2/(7.)))) and so on

G.C.

Dear Naveen,

without more context I could think of one exact rewriting which can be evaluated more accurately on the computer and in case you calculate integrals a simplification.

1) a rewriting: 

TANH(t) = [1 - exp(-2t)]/[1 + exp(-2t)]  for  t>=0

and

TANH(t) = [exp(2t) - 1]/[exp(2t) + 1] for t

Thanks all for giving your suggestions. I am looking for a simpler function which may contain a linear function of non-linear terms (t^p). The approximation should have at least 3 decimal point accuracy. Kindly suggest if any. Thanks in advance...

Shakeel Ahmad Malik