Symbolic points in an elliptic curve over $\mathbb{Q}$ of the form $(u/e^{2},v/e^{3})$

Given an elliptic curve $$ y^2 = x^3 + ax + b $$ and points on the curve$\,P = (u_1/e_1^2, v_1/e_1^3),\,$ $\,Q = (u_2/e_2^2,v_2/e_2^3),\,$ then $\,R := P+Q = (u_3/e_3^3, v_3/e_3^3)\, $ where $$u_3 = e_2^2 e_1^4 u_1 u_2^2+e_2^4 e_1^2 u_1^2 u_2-e_2^6 u_1^3+e_1^6 v_2^2-2 e_2^3 e_1^3 v_1 v_2+e_2^6 v_1^2 -e_1^6u_2^3, $$ $$v_3 = e_1^9 u_2^3 v_2 - 2 e_2^3 e_1^6 u_2^3 v_1 - 3 e_2^4 e_1^5 u_1^2 u_2 v_2 + 3e_2^5 e_1^4 u_1 u_2^2 v_1 + 2 e_2^6 e_1^3 u_1^3 v_2-e_2^9 u_1^3 v_1 - e_1^9v_2^3 + 3 e_2^3 e_1^6 v_1 v_2^2 - 3e_2^6 e_1^3 v_1^2 v_2 + e_2^9 v_1^3, $$ $$e_3 = e_1 e_2 (e_1^2 u_2-e_2^2 u_1).$$

If $\,P = (u/e^2, v/e^3),\,$ then $\,2P = (u_2/e_2^2,v_2/e_2^3),\,$ where $$ u_2 = a^2 e^8+6 a e^4 u^2+9 u^4-8 u v^2,\qquad e_2 = 2 e v, $$ $$ v_2 = -a^3 e^{12}-9 a^2 e^8 u^2-27 a e^4 u^4+12 a e^4 u v^2-27 u^6+36 u^3 v^2-8 v^4. $$ I found the formulas I needed on a web page at Project Nayuki.