Minimax polynomial approximations (§3.11(i)) for
and 
in terms of 
with
can be found in Abramowitz and Stegun (1964, §17.3) with maximum
absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the
same type for and for 
are
given in Cody (1965a) with maximum absolute errors ranging from
4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series
expansions (§3.11(ii)) with maximum precision 25D.
Approximations for Legendre’s complete or incomplete integrals of all three
kinds, derived by Padé approximation of the square root in the integrand,
are given in Luke (1968, 1970). They are valid
over parts of the complex 
and 
planes. The accuracy is controlled by
the number of terms retained in the approximation; for real variables the
number of significant figures appears to be roughly twice the number of terms
retained, perhaps even for near 
with the improvements made in the
1970 reference.
