My guess is that fake( exp(x) - (x^3 / 5) ) is simply :

exp(x) - Constant.

But I have not tried anything yet.

I assume the best possible approximation ( by using fake - for exp(x) - ( x^3 / 5 ) - ) is approximately exp(x) - a x^b - constant.

Where b is between 0 and 1.

Also a possibility is with q_i very small :

1 + q_1 x + q_2 x^2 + q_3 x^3 + x^4/4! + x^5/5! + ...

or

0 + q_1 x + q_2 x^2 + q_3 x^3 + x^4/4! + x^5/5! + ...

I wonder.

It seems 0 + q_1 x + q_2 x^2 + q_3 x^3 + x^4/4! + x^5/5! + ...

cannot be improved.

( because O(x^3) cannot be given by O(x^(3+eps)) )

Therefore it would be a good (benchmark) test for our fake algorithms I guess.

regards

tommy1729

exp(x) - Constant.

But I have not tried anything yet.

I assume the best possible approximation ( by using fake - for exp(x) - ( x^3 / 5 ) - ) is approximately exp(x) - a x^b - constant.

Where b is between 0 and 1.

Also a possibility is with q_i very small :

1 + q_1 x + q_2 x^2 + q_3 x^3 + x^4/4! + x^5/5! + ...

or

0 + q_1 x + q_2 x^2 + q_3 x^3 + x^4/4! + x^5/5! + ...

I wonder.

It seems 0 + q_1 x + q_2 x^2 + q_3 x^3 + x^4/4! + x^5/5! + ...

cannot be improved.

( because O(x^3) cannot be given by O(x^(3+eps)) )

Therefore it would be a good (benchmark) test for our fake algorithms I guess.

regards

tommy1729