nx_trig.nx

nx_trig.nx -- cos / sin primitives in Q10 fixed point. Bits-up math primitive sibling of nx_isqrt.nx + nx_exp.nx + nx_root.nx. Provides the canonical trigonometric kernel everything else composes against -- RoPE position embeddings (Su 2021), sinusoidal position encoding (Vaswani 2017), DSP filters, rotation matrices, 2D/3D graphics, signal-processing FFT (when shipped), etc. ===== Domain choice: turn-fraction not radians =================== Storing angle in radians (Q10) forces 2pi-range-reduction on every call, which is awkward in i64 because 2pi = 6283 Q10 doesn't divide any power-of-2 cleanly. Storing as TURN FRACTION (1 turn = 1 full circle = 2pi rad) makes range reduction a single mod-Q10 operation and aligns directly with the 4-quadrant symmetry. turn_q10 in [0, NX_TRIG_TURN) -- one Q10 unit per (2pi/1024) rad NX_TRIG_TURN = 1024 = a full circle in Q10 Per-quadrant breakdown: [0, NX_TRIG_TURN/4) : cos > 0, sin > 0 [TURN/4, TURN/2) : cos < 0, sin > 0 [TURN/2, 3TURN/4) : cos < 0, sin < 0 [3TURN/4, TURN) : cos > 0, sin < 0 Lookup table covers the first quadrant only (256 entries for 1/4 turn at NX_TRIG_TURN=1024). Other three quadrants derived by symmetry without any table extension. Conversion: radians-to-turns and back are exposed for callers who prefer radians input. ===== Quality envelope (honest) ================================== 256-entry table + linear interpolation over a quarter turn gives max relative error ~0.5% across (0, pi/2) -- below substrate Q10-precision floor. Identities `sin^2 + cos^2 = Q10^2` hold to within +/- 8 Q10 units. Sufficient for RoPE / position encoding / rotation matrices; for FFT-class precision a Padé approximant (Q14 or Q20) upgrade is queued.