Time of the seasons

[Giuseppe Donnini]

The discrepancy in the last digit between Meeus’ result and yours stems from the fact that you calculated the polynomial directly:

$$
JDE_{0}=\,2\,451\,716.56767 + 365\,241.62603\cdot Y + 0.00325\cdot Y^2 + 0.00888\cdot Y^3 - 0.00030\cdot Y^4
$$

instead of using Horner’s method, as Meeus himself recommends (see the general discussion on pp. 10-11):

$$
\Leftrightarrow \;2\,451\,716.56767 + Y\cdot ( \,365\,241.62603 + Y\cdot ( \,0.00325 + Y\cdot ( \,0.00888 - Y\cdot 0.00030\,) \,) \,)
$$

In general, the latter is both faster and – most importantly – more accurate:

1. If n is the degree of the polynomial, direct evaluation would require n(n+1)/2 multiplications, while Horner’s method only requires n. The larger n, the larger the speed gain.
    
2. Fewer operations automatically translate into less accumulation of roundoff errors.