  We describe a strategy for rigorous arbitrary-precision evaluation of
Legendre polynomials on the unit interval and its application in the generation
of Gauss-Legendre quadrature rules. Our focus is on making the evaluation
practical for a wide range of realistic parameters, corresponding to the
requirements of numerical integration to an accuracy of about 100 to 100 000
bits. Our algorithm combines the summation by rectangular splitting of several
types of expansions in terms of hypergeometric series with a fixed-point
implementation of Bonnet's three-term recurrence relation. We then compute
rigorous enclosures of the Gauss-Legendre nodes and weights using the interval
Newton method. We provide rigorous error bounds for all steps of the algorithm.
The approach is validated by an implementation in the Arb library, which
achieves order-of-magnitude speedups over previous code for computing
Gauss-Legendre rules with simultaneous high degree and precision.
