1 /* 2 * Double-precision vector tan(x) function. 3 * 4 * Copyright (c) 2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "v_math.h" 9 #include "poly_advsimd_f64.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 static const struct data 14 { 15 float64x2_t poly[9]; 16 float64x2_t half_pi, two_over_pi, shift; 17 #if !WANT_SIMD_EXCEPT 18 float64x2_t range_val; 19 #endif 20 } data = { 21 /* Coefficients generated using FPMinimax. */ 22 .poly = { V2 (0x1.5555555555556p-2), V2 (0x1.1111111110a63p-3), 23 V2 (0x1.ba1ba1bb46414p-5), V2 (0x1.664f47e5b5445p-6), 24 V2 (0x1.226e5e5ecdfa3p-7), V2 (0x1.d6c7ddbf87047p-9), 25 V2 (0x1.7ea75d05b583ep-10), V2 (0x1.289f22964a03cp-11), 26 V2 (0x1.4e4fd14147622p-12) }, 27 .half_pi = { 0x1.921fb54442d18p0, 0x1.1a62633145c07p-54 }, 28 .two_over_pi = V2 (0x1.45f306dc9c883p-1), 29 .shift = V2 (0x1.8p52), 30 #if !WANT_SIMD_EXCEPT 31 .range_val = V2 (0x1p23), 32 #endif 33 }; 34 35 #define RangeVal 0x4160000000000000 /* asuint64(0x1p23). */ 36 #define TinyBound 0x3e50000000000000 /* asuint64(2^-26). */ 37 #define Thresh 0x310000000000000 /* RangeVal - TinyBound. */ 38 39 /* Special cases (fall back to scalar calls). */ 40 static float64x2_t VPCS_ATTR NOINLINE 41 special_case (float64x2_t x) 42 { 43 return v_call_f64 (tan, x, x, v_u64 (-1)); 44 } 45 46 /* Vector approximation for double-precision tan. 47 Maximum measured error is 3.48 ULP: 48 _ZGVnN2v_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37 49 want -0x1.f6ccd8ecf7deap+37. */ 50 float64x2_t VPCS_ATTR V_NAME_D1 (tan) (float64x2_t x) 51 { 52 const struct data *dat = ptr_barrier (&data); 53 /* Our argument reduction cannot calculate q with sufficient accuracy for 54 very large inputs. Fall back to scalar routine for all lanes if any are 55 too large, or Inf/NaN. If fenv exceptions are expected, also fall back for 56 tiny input to avoid underflow. */ 57 #if WANT_SIMD_EXCEPT 58 uint64x2_t iax = vreinterpretq_u64_f64 (vabsq_f64 (x)); 59 /* iax - tiny_bound > range_val - tiny_bound. */ 60 uint64x2_t special 61 = vcgtq_u64 (vsubq_u64 (iax, v_u64 (TinyBound)), v_u64 (Thresh)); 62 if (unlikely (v_any_u64 (special))) 63 return special_case (x); 64 #endif 65 66 /* q = nearest integer to 2 * x / pi. */ 67 float64x2_t q 68 = vsubq_f64 (vfmaq_f64 (dat->shift, x, dat->two_over_pi), dat->shift); 69 int64x2_t qi = vcvtq_s64_f64 (q); 70 71 /* Use q to reduce x to r in [-pi/4, pi/4], by: 72 r = x - q * pi/2, in extended precision. */ 73 float64x2_t r = x; 74 r = vfmsq_laneq_f64 (r, q, dat->half_pi, 0); 75 r = vfmsq_laneq_f64 (r, q, dat->half_pi, 1); 76 /* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle 77 formula. */ 78 r = vmulq_n_f64 (r, 0.5); 79 80 /* Approximate tan(r) using order 8 polynomial. 81 tan(x) is odd, so polynomial has the form: 82 tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ... 83 Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ... 84 Then compute the approximation by: 85 tan(r) ~= r + r^3 * (C0 + r^2 * P(r)). */ 86 float64x2_t r2 = vmulq_f64 (r, r), r4 = vmulq_f64 (r2, r2), 87 r8 = vmulq_f64 (r4, r4); 88 /* Offset coefficients to evaluate from C1 onwards. */ 89 float64x2_t p = v_estrin_7_f64 (r2, r4, r8, dat->poly + 1); 90 p = vfmaq_f64 (dat->poly[0], p, r2); 91 p = vfmaq_f64 (r, r2, vmulq_f64 (p, r)); 92 93 /* Recombination uses double-angle formula: 94 tan(2x) = 2 * tan(x) / (1 - (tan(x))^2) 95 and reciprocity around pi/2: 96 tan(x) = 1 / (tan(pi/2 - x)) 97 to assemble result using change-of-sign and conditional selection of 98 numerator/denominator, dependent on odd/even-ness of q (hence quadrant). 99 */ 100 float64x2_t n = vfmaq_f64 (v_f64 (-1), p, p); 101 float64x2_t d = vaddq_f64 (p, p); 102 103 uint64x2_t no_recip = vtstq_u64 (vreinterpretq_u64_s64 (qi), v_u64 (1)); 104 105 #if !WANT_SIMD_EXCEPT 106 uint64x2_t special = vcageq_f64 (x, dat->range_val); 107 if (unlikely (v_any_u64 (special))) 108 return special_case (x); 109 #endif 110 111 return vdivq_f64 (vbslq_f64 (no_recip, n, vnegq_f64 (d)), 112 vbslq_f64 (no_recip, d, n)); 113 } 114 115 PL_SIG (V, D, 1, tan, -3.1, 3.1) 116 PL_TEST_ULP (V_NAME_D1 (tan), 2.99) 117 PL_TEST_EXPECT_FENV (V_NAME_D1 (tan), WANT_SIMD_EXCEPT) 118 PL_TEST_SYM_INTERVAL (V_NAME_D1 (tan), 0, TinyBound, 5000) 119 PL_TEST_SYM_INTERVAL (V_NAME_D1 (tan), TinyBound, RangeVal, 100000) 120 PL_TEST_SYM_INTERVAL (V_NAME_D1 (tan), RangeVal, inf, 5000) 121