1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision vector tan(x) function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2023-2024, Arm Limited. 5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6*f3087befSAndrew Turner */ 7*f3087befSAndrew Turner 8*f3087befSAndrew Turner #include "v_math.h" 9*f3087befSAndrew Turner #include "v_poly_f64.h" 10*f3087befSAndrew Turner #include "test_sig.h" 11*f3087befSAndrew Turner #include "test_defs.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner static const struct data 14*f3087befSAndrew Turner { 15*f3087befSAndrew Turner float64x2_t poly[9]; 16*f3087befSAndrew Turner double half_pi[2]; 17*f3087befSAndrew Turner float64x2_t two_over_pi, shift; 18*f3087befSAndrew Turner #if !WANT_SIMD_EXCEPT 19*f3087befSAndrew Turner float64x2_t range_val; 20*f3087befSAndrew Turner #endif 21*f3087befSAndrew Turner } data = { 22*f3087befSAndrew Turner /* Coefficients generated using FPMinimax. */ 23*f3087befSAndrew Turner .poly = { V2 (0x1.5555555555556p-2), V2 (0x1.1111111110a63p-3), 24*f3087befSAndrew Turner V2 (0x1.ba1ba1bb46414p-5), V2 (0x1.664f47e5b5445p-6), 25*f3087befSAndrew Turner V2 (0x1.226e5e5ecdfa3p-7), V2 (0x1.d6c7ddbf87047p-9), 26*f3087befSAndrew Turner V2 (0x1.7ea75d05b583ep-10), V2 (0x1.289f22964a03cp-11), 27*f3087befSAndrew Turner V2 (0x1.4e4fd14147622p-12) }, 28*f3087befSAndrew Turner .half_pi = { 0x1.921fb54442d18p0, 0x1.1a62633145c07p-54 }, 29*f3087befSAndrew Turner .two_over_pi = V2 (0x1.45f306dc9c883p-1), 30*f3087befSAndrew Turner .shift = V2 (0x1.8p52), 31*f3087befSAndrew Turner #if !WANT_SIMD_EXCEPT 32*f3087befSAndrew Turner .range_val = V2 (0x1p23), 33*f3087befSAndrew Turner #endif 34*f3087befSAndrew Turner }; 35*f3087befSAndrew Turner 36*f3087befSAndrew Turner #define RangeVal 0x4160000000000000 /* asuint64(0x1p23). */ 37*f3087befSAndrew Turner #define TinyBound 0x3e50000000000000 /* asuint64(2^-26). */ 38*f3087befSAndrew Turner #define Thresh 0x310000000000000 /* RangeVal - TinyBound. */ 39*f3087befSAndrew Turner 40*f3087befSAndrew Turner /* Special cases (fall back to scalar calls). */ 41*f3087befSAndrew Turner static float64x2_t VPCS_ATTR NOINLINE 42*f3087befSAndrew Turner special_case (float64x2_t x) 43*f3087befSAndrew Turner { 44*f3087befSAndrew Turner return v_call_f64 (tan, x, x, v_u64 (-1)); 45*f3087befSAndrew Turner } 46*f3087befSAndrew Turner 47*f3087befSAndrew Turner /* Vector approximation for double-precision tan. 48*f3087befSAndrew Turner Maximum measured error is 3.48 ULP: 49*f3087befSAndrew Turner _ZGVnN2v_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37 50*f3087befSAndrew Turner want -0x1.f6ccd8ecf7deap+37. */ 51*f3087befSAndrew Turner float64x2_t VPCS_ATTR V_NAME_D1 (tan) (float64x2_t x) 52*f3087befSAndrew Turner { 53*f3087befSAndrew Turner const struct data *dat = ptr_barrier (&data); 54*f3087befSAndrew Turner /* Our argument reduction cannot calculate q with sufficient accuracy for 55*f3087befSAndrew Turner very large inputs. Fall back to scalar routine for all lanes if any are 56*f3087befSAndrew Turner too large, or Inf/NaN. If fenv exceptions are expected, also fall back for 57*f3087befSAndrew Turner tiny input to avoid underflow. */ 58*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT 59*f3087befSAndrew Turner uint64x2_t iax = vreinterpretq_u64_f64 (vabsq_f64 (x)); 60*f3087befSAndrew Turner /* iax - tiny_bound > range_val - tiny_bound. */ 61*f3087befSAndrew Turner uint64x2_t special 62*f3087befSAndrew Turner = vcgtq_u64 (vsubq_u64 (iax, v_u64 (TinyBound)), v_u64 (Thresh)); 63*f3087befSAndrew Turner if (unlikely (v_any_u64 (special))) 64*f3087befSAndrew Turner return special_case (x); 65*f3087befSAndrew Turner #endif 66*f3087befSAndrew Turner 67*f3087befSAndrew Turner /* q = nearest integer to 2 * x / pi. */ 68*f3087befSAndrew Turner float64x2_t q 69*f3087befSAndrew Turner = vsubq_f64 (vfmaq_f64 (dat->shift, x, dat->two_over_pi), dat->shift); 70*f3087befSAndrew Turner int64x2_t qi = vcvtq_s64_f64 (q); 71*f3087befSAndrew Turner 72*f3087befSAndrew Turner /* Use q to reduce x to r in [-pi/4, pi/4], by: 73*f3087befSAndrew Turner r = x - q * pi/2, in extended precision. */ 74*f3087befSAndrew Turner float64x2_t r = x; 75*f3087befSAndrew Turner float64x2_t half_pi = vld1q_f64 (dat->half_pi); 76*f3087befSAndrew Turner r = vfmsq_laneq_f64 (r, q, half_pi, 0); 77*f3087befSAndrew Turner r = vfmsq_laneq_f64 (r, q, half_pi, 1); 78*f3087befSAndrew Turner /* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle 79*f3087befSAndrew Turner formula. */ 80*f3087befSAndrew Turner r = vmulq_n_f64 (r, 0.5); 81*f3087befSAndrew Turner 82*f3087befSAndrew Turner /* Approximate tan(r) using order 8 polynomial. 83*f3087befSAndrew Turner tan(x) is odd, so polynomial has the form: 84*f3087befSAndrew Turner tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ... 85*f3087befSAndrew Turner Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ... 86*f3087befSAndrew Turner Then compute the approximation by: 87*f3087befSAndrew Turner tan(r) ~= r + r^3 * (C0 + r^2 * P(r)). */ 88*f3087befSAndrew Turner float64x2_t r2 = vmulq_f64 (r, r), r4 = vmulq_f64 (r2, r2), 89*f3087befSAndrew Turner r8 = vmulq_f64 (r4, r4); 90*f3087befSAndrew Turner /* Offset coefficients to evaluate from C1 onwards. */ 91*f3087befSAndrew Turner float64x2_t p = v_estrin_7_f64 (r2, r4, r8, dat->poly + 1); 92*f3087befSAndrew Turner p = vfmaq_f64 (dat->poly[0], p, r2); 93*f3087befSAndrew Turner p = vfmaq_f64 (r, r2, vmulq_f64 (p, r)); 94*f3087befSAndrew Turner 95*f3087befSAndrew Turner /* Recombination uses double-angle formula: 96*f3087befSAndrew Turner tan(2x) = 2 * tan(x) / (1 - (tan(x))^2) 97*f3087befSAndrew Turner and reciprocity around pi/2: 98*f3087befSAndrew Turner tan(x) = 1 / (tan(pi/2 - x)) 99*f3087befSAndrew Turner to assemble result using change-of-sign and conditional selection of 100*f3087befSAndrew Turner numerator/denominator, dependent on odd/even-ness of q (hence quadrant). 101*f3087befSAndrew Turner */ 102*f3087befSAndrew Turner float64x2_t n = vfmaq_f64 (v_f64 (-1), p, p); 103*f3087befSAndrew Turner float64x2_t d = vaddq_f64 (p, p); 104*f3087befSAndrew Turner 105*f3087befSAndrew Turner uint64x2_t no_recip = vtstq_u64 (vreinterpretq_u64_s64 (qi), v_u64 (1)); 106*f3087befSAndrew Turner 107*f3087befSAndrew Turner #if !WANT_SIMD_EXCEPT 108*f3087befSAndrew Turner uint64x2_t special = vcageq_f64 (x, dat->range_val); 109*f3087befSAndrew Turner if (unlikely (v_any_u64 (special))) 110*f3087befSAndrew Turner return special_case (x); 111*f3087befSAndrew Turner #endif 112*f3087befSAndrew Turner 113*f3087befSAndrew Turner return vdivq_f64 (vbslq_f64 (no_recip, n, vnegq_f64 (d)), 114*f3087befSAndrew Turner vbslq_f64 (no_recip, d, n)); 115*f3087befSAndrew Turner } 116*f3087befSAndrew Turner 117*f3087befSAndrew Turner TEST_SIG (V, D, 1, tan, -3.1, 3.1) 118*f3087befSAndrew Turner TEST_ULP (V_NAME_D1 (tan), 2.99) 119*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_D1 (tan), WANT_SIMD_EXCEPT) 120*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_D1 (tan), 0, TinyBound, 5000) 121*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_D1 (tan), TinyBound, RangeVal, 100000) 122*f3087befSAndrew Turner TEST_SYM_INTERVAL (V_NAME_D1 (tan), RangeVal, inf, 5000) 123