mp_real_partitions_hrr, _mp_real_partitions_hrr

mp_real_partitions_hrr reads $p(n)$ from a table for $n < 128$, uses the pentagonal recurrence while $p(n)$ fits a word ($n < 417$ on 64-bit machines), and otherwise the Hardy–Ramanujan–Rademacher formula [Rademacher1937] [Joh2012]

with $A_k(n)$ from arith_hrr_expsum_factored, $N$ and the bound for $R_N$ from partitions_rademacher_bound, and each term at the precision it contributes at. Terms of at most about 200 bits are evaluated in batches of dfloat balls, the others in mp_real balls, with $\exp(z_k)$ obtained as the $k$-th root of $\exp(z_1)$ (computed by mp_real_exp_notab_squaring or, at huge precision, mp_real_exp_agm). The resulting ball is resolved by mp_real_unique_integer (Integers from balls).

With several threads (flint_set_num_threads, from $n = 10^8$) the constants are computed once with the whole thread budget, and the terms are summed in ranges of $k$ of about equal cost, each into its own accumulator. The peak memory is about that of computing $\pi$ to the same precision.

_mp_real_partitions_hrr uses the formula for every $n \ge 2$; flags is a combination of MP_REAL_PARTITIONS_NO_DFLOAT (all terms in mp_real) and MP_REAL_PARTITIONS_TEST (the rarely taken paths at any $n$), for testing. partitions_fmpz_fmpz in partitions.h calls these functions. The method is described in Computing the partition function.

@misc{flint-doc-mp-real-mp-real-partitions-hrr,
  author       = {{The FLINT team}},
  title        = {{mp\_real\_partitions\_hrr, \_mp\_real\_partitions\_hrr (mp\_real.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/mp_real/mp_real_partitions_hrr.html},
  note         = {Generated 2026-11-02}
}