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You seem to expect that f(x).taylor(x,0 3) is equal to f(x).abs().taylor(x,0,3).
There's nothing allowing such an expectation. Simple illustrative counter-example :
sage: with assuming(x, "real"):(cos(x)+I*sin(x)).exponentialize().simplify().taylor(x,0,5)
1/120*I*x^5 + 1/24*x^4 - 1/6*I*x^3 - 1/2*x^2 + I*x + 1
sage: with assuming(x, "real"):(cos(x)+I*sin(x)).abs().exponentialize().simplify().taylor(x,0,5)
1
HTH,
