Find the largest x for which x^b+a = a

Question

Stability (Numerical analysis)

Trying to apply the answer I saw in this question, a+x=a worked just fine with a+eps(a)/2. Suppose we have x^b+a=a, where b is a small integer, say 3 and a=2000. Then a+(eps(a))^3 or a+(eps(a)/2)^3 will always return number a. Can someone help with the measurement of x? Any way, even different from eps will do just fine.

p.s. 1938+(eps(1938)/0.00000000469)^3 is the last number that returns ans = 1.9380e+003. 1938+(eps(1938)/0.0000000047)^3 returns a=1938. Does that have to do with anything?

Answer 1

x = (eps(a)/2).^(1/(b-eps(a)/2))

if b = 3,

(eps(1938)/2).^(1/(3-eps(1938)/2)) > eps(1938)/0.0000000047