Why do you use floating-point numbers in the first place?!
I mean, are you trying to use these values, or something similar?
phi, phi, phi, phi, phi, phi
:= sqrt(2), sqrt(2), (4*t-1)*sqrt(6), (4*t-3)*sqrt(6),
If so then why not use exact rationals for alpha, lambda, etc. And why not subsequently construct the eqs[i] from the X.Y,H,etc and their derivatives?
Maybe you'd be able to get an exact root, which you could approximate quickly and with less accumulated roundoff error.
Maybe you'd even be able to get an exact symbolic formula for a generic form of the root for general k and M. (I can't tell, because you've already done the dubious step of cut and pasting just to get the formulas used used to construct eq[i]. And the formulation has been obscured...).