FLINT3.7.0-dev
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_mp_real_sin_cos_pi_ui_div_ui, _mp_real_sin_cos_pi_ui_div_ui_tune

Synopsis#include <flint/mp_real.h>
void _mp_real_sin_cos_pi_ui_div_ui(nn_ptr ys, nn_ptr yc, ulong * err, ulong p, ulong q, slong n)source
void _mp_real_sin_cos_pi_ui_div_ui_tune(nn_ptr ys, nn_ptr yc, ulong * err, ulong p, ulong q, slong n, int alg, int r)source

Set (ys,n+1)(y_s, n + 1) and (yc,n+1)(y_c, n + 1) (either may be NULL) to sin⁡(πp/q)\sin(\pi p/q) and cos⁡(πp/q)\cos(\pi p/q) for q≥1q \ge 1 and 0≤p≤q/20 \le p \le q/2. err is at most 4 for the closed forms and the Chebyshev iteration and the kernel's bound plus 2 (at most 6r+1306r + 130, r≤768r \le 768) when the sine and cosine kernel is used.

After reduction of the fraction, q≤2q \le 2 gives exact values, q=3,4,6q = 3, 4, 6 quadratic irrationals from _mp_real_rsqrt_ui_newton, and q=5,8,10,12q = 5, 8, 10, 12 closed forms with square roots from about 12 limbs (24 for q=8q = 8). For denominators up to a precision-dependent bit length (none below 48 limbs, 12 bits at 512 limbs, 48 bits at 65536 limbs, any beyond) the value is refined as a root of a Chebyshev polynomial by a high-order iteration costing O(M(n)log⁡q)O(M(n) \log q); otherwise the kernel of mp_real_sin_cos_bits is applied to πp/q\pi p/q. The _tune version takes the algorithm (alg = 0 tuned, 1 kernel, 2 Chebyshev iteration, 3 closed forms) and the order of the iteration (r = 0 for the default 12, at most 16). See Sines of rational multiples of π.

Cite this page

The FLINT team, “_mp_real_sin_cos_pi_ui_div_ui, _mp_real_sin_cos_pi_ui_div_ui_tune (mp_real.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/mp_real/_mp_real_sin_cos_pi_ui_div_ui.html (generated 2026-11-02).