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https://twitter.com/IneffectiveMath/status/1635296761672769543

At first we could explore:

f(0) = 0
g(0) = 0
p=0
E_c(0) = g(f(0)+c)=g(0+c) = g(c)
g(e) = E_e(0) = E_c(0)*E_d(q)

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Next we could explore:

c = 0
d = 0
E_0(p)E_0(q) = E_e(pq)
g(f(p))g(f(q)) = E_e(pq) = g(f(pq) + e)

Here, e=0 works.