best approximation for double numbers

I have the following function which has an asymptote for y = 1 :

syms f(x)
syms x
f(x) = 1 / (1 + exp(-((x + 8.4730) / 10 )));

So, so the function should have a value < 1 for all x.

Also using "format long", of course, from a certain value of x onwards, the result of the function is approximated to 1.

format long
double(f(400))  % ans = 1

Is there a way to get an approximation to the exact value for even larger x? Or should I settle for this approximation?

Answer 1

[VBBV]

format long

syms f(x)
syms x
f(x) = 1 / (1 + exp(-((x + 8.4730) / 10 )))

f(x) = 

vpa(f(400),100)

ans = 

0.9999999999999999981792806433707777978855866136633285567221016451791545277454316141363423975013892837

[VBBV]

Try using vpa for large values of x

[Walter Roberson]

syms f(x)
syms x
f(x) = 1 / (1 + exp(-((x + 8.4730) / 10 )))

f(x) = 

f1 = simplify(expand(1-f))

f1(x) = 

double(f1(400))

ans = 1.8207e-18

fplot(f1,[500 600])

f1n = matlabFunction(f1)

f1n = function_handle with value:
    @(x)1.0./(exp(x./1.0e+1+8.473e-1)+1.0)

fplot(f1n, [500 600])

f1 gives you an idea of how quickly the value approaches 1, by showing you how quickly the difference between 1 and f falls. f1n shows that a numeric approximation (instead of a symbolic) of 1-f is still not bad at all in this kind of range.