source 2006 Doche--Icart--Kohel "Efficient scalar multiplication by isogeny decompositions", page 198, bottom display, plus correction from 2007 Bernstein--Birkner--Lange--Peters to incorporate the missing -3a term
parameter a3
assume a3 = 3 a
compute A = Y1 Z1
compute Z3 = 2 A
compute B = 4 Y1^2 X1
compute C = B+6 a A^2
compute ZZ3 = 4 A^2
compute D = 3 X1^2
compute E = D+6 a ZZ1(ZZ1+X1)
compute X3 = E^2-2 B-a3 ZZ3
compute Y3 = E(B-X3)-8 Y1^4