nx_mathspec_tan.nx

nx_mathspec_tan.nx -- SOVEREIGN SPEC EMITTER for the TAN kernel: every constant from the 120-bit bigfloat (trig + tan oracle organs). No python, no minimax: the small-residual polynomial uses the EXACT tangent-number Taylor family tan d = d + sum_{n>=2} T_n d^(2n-1)/(2n-1)! with the integer tangent numbers T_2..T_8 = 2 16 272 7936 353792 22368256 1903757312 (OEIS A000182, mathematical integers like k!); each coefficient divides by 2..(2n-1) one factor at a time (bf_div_small bound). |d| <= 1/16 -> 7 correction terms truncate below 2^-57 relative. Kernel scheme (accurate-table + dd): pi/2 Cody-Waite 3-part reduce (same 21-zeroed HI/MID construction as the sincos spec) -> r dd; anchors a_j = j/8 (j = 0..6 covers [0, pi/4]); tan(a+d) = T + t*sec2(a)/(1 - T*t) with T and sec2 carried hi/lo from this spec and the quotient refined in dd; odd quadrants reconstruct -1/tan via a dd reciprocal. Layout: c[0]=one c[1]=half c[2]=2/pi c[3]=HI c[4]=MID c[5]=LO c[6]=eight c[7..13] anchors j/8 c[14..20] T_hi[j] = fl(tan(j/8)) c[21..27] T_lo[j] (signed) c[28..34] S_hi[j] = fl(1+tan^2(j/8)) c[35..41] S_lo[j] (signed) c[42..48] Q coeffs hi-first (T_8.. down to T_2) license_tier: ORIGINAL