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I tried to fit the following data [blue dots] by the following function [red],

$$f(x,a,b,c,d,e)text:=d-fracc1+ exp left(frac(x-a)^ebright)$$

Values of fitted parameters can be seen in the figure.

The number of unknown parameters can be reduced by fixing $a$ and $e$ (or some other as well).

Now, Ques is: I am looking for a better algebraic simple fitting function.

Any idea, please ?

P.S.- Sorry, If this question is not eligible to post here!!

The inspection of the data shows that two parts are linear with an high accuracy.

Obviously the points $(x_k,y_k)$ with $77leq k leq 100$ are exactly located on the horizontal line $y= 5.92608$ .

The points $(x_k,y_k)$ with $1leq k leq 10$ are very closely located on a straight line $y= a+bx$ :

For the other points $11leq k leq 76$ on can find various functions with a good fit.

For example this very simple form : $quad ysimeq fraca+bx+cx^21+Bx$

COMMENT :

With straight segments in a curve one usually chose a piecewise function to fit accurately the data.

In this case, an example of piecewise function is : $quad ysimeqbegincases
alpha +beta x && x<x_m\
fraca+bx+cx^21+Bx && x_mleq xleq x_M\
C && x_M<x
endcasesquad$ ( Next figure).

Or with the Heaviside step function :

$$ysimeq (alpha +beta x)left(1-H(x-x_m) right)+fraca+bx+cx^21+BxH(x-x_m)left(1-H(x-x_M)right)+C:H(x-x_M)$$

With a continuous function differentiable everywhere one canot expect an accurate fitting :

Such functions will not fit accurately the straight parts. Moreover, they will introduce some inevitable deviations on the curved part, which could be avoid with a piecewise function.

This is what we can see with the function that you show. By the way your function appears as one of the best. Of course it is always possible to improve with more adjustable parameters in the function.

Note : If the data comes from real experiments, it is always better to use a function deduced from theory and modeling of the physical phenomenon, even if the fitting is worse than with a purely mathematical choice of function, because it is of interest that the parameters have a real physical meaning.

Since you are looking for an algebraic function, you may try this:

$$f(x)=d-c left(1+fracx-abright)^-e$$

Which gives the following fit: See here the plot of fitted function with the data you provided.

Fitted parameters are a=2, b=3.34, c=7.94, d=7.81, e=0.5.