1/* Double-precision vector (Advanced SIMD) tan function
2
3 Copyright (C) 2023-2024 Free Software Foundation, Inc.
4 This file is part of the GNU C Library.
5
6 The GNU C Library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
10
11 The GNU C Library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
15
16 You should have received a copy of the GNU Lesser General Public
17 License along with the GNU C Library; if not, see
18 <https://www.gnu.org/licenses/>. */
19
20#include "v_math.h"
21#include "poly_advsimd_f64.h"
22
23static const struct data
24{
25 float64x2_t poly[9];
26 float64x2_t half_pi, two_over_pi, shift;
27#if !WANT_SIMD_EXCEPT
28 float64x2_t range_val;
29#endif
30} data = {
31 /* Coefficients generated using FPMinimax. */
32 .poly = { V2 (0x1.5555555555556p-2), V2 (0x1.1111111110a63p-3),
33 V2 (0x1.ba1ba1bb46414p-5), V2 (0x1.664f47e5b5445p-6),
34 V2 (0x1.226e5e5ecdfa3p-7), V2 (0x1.d6c7ddbf87047p-9),
35 V2 (0x1.7ea75d05b583ep-10), V2 (0x1.289f22964a03cp-11),
36 V2 (0x1.4e4fd14147622p-12) },
37 .half_pi = { 0x1.921fb54442d18p0, 0x1.1a62633145c07p-54 },
38 .two_over_pi = V2 (0x1.45f306dc9c883p-1),
39 .shift = V2 (0x1.8p52),
40#if !WANT_SIMD_EXCEPT
41 .range_val = V2 (0x1p23),
42#endif
43};
44
45#define RangeVal 0x4160000000000000 /* asuint64(0x1p23). */
46#define TinyBound 0x3e50000000000000 /* asuint64(2^-26). */
47#define Thresh 0x310000000000000 /* RangeVal - TinyBound. */
48
49/* Special cases (fall back to scalar calls). */
50static float64x2_t VPCS_ATTR NOINLINE
51special_case (float64x2_t x)
52{
53 return v_call_f64 (tan, x, x, v_u64 (x: -1));
54}
55
56/* Vector approximation for double-precision tan.
57 Maximum measured error is 3.48 ULP:
58 _ZGVnN2v_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37
59 want -0x1.f6ccd8ecf7deap+37. */
60float64x2_t VPCS_ATTR V_NAME_D1 (tan) (float64x2_t x)
61{
62 const struct data *dat = ptr_barrier (&data);
63 /* Our argument reduction cannot calculate q with sufficient accuracy for
64 very large inputs. Fall back to scalar routine for all lanes if any are
65 too large, or Inf/NaN. If fenv exceptions are expected, also fall back for
66 tiny input to avoid underflow. */
67#if WANT_SIMD_EXCEPT
68 uint64x2_t iax = vreinterpretq_u64_f64 (vabsq_f64 (x));
69 /* iax - tiny_bound > range_val - tiny_bound. */
70 uint64x2_t special
71 = vcgtq_u64 (vsubq_u64 (iax, v_u64 (TinyBound)), v_u64 (Thresh));
72 if (__glibc_unlikely (v_any_u64 (special)))
73 return special_case (x);
74#endif
75
76 /* q = nearest integer to 2 * x / pi. */
77 float64x2_t q
78 = vsubq_f64 (vfmaq_f64 (dat->shift, x, dat->two_over_pi), dat->shift);
79 int64x2_t qi = vcvtq_s64_f64 (q);
80
81 /* Use q to reduce x to r in [-pi/4, pi/4], by:
82 r = x - q * pi/2, in extended precision. */
83 float64x2_t r = x;
84 r = vfmsq_laneq_f64 (r, q, dat->half_pi, 0);
85 r = vfmsq_laneq_f64 (r, q, dat->half_pi, 1);
86 /* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle
87 formula. */
88 r = vmulq_n_f64 (r, 0.5);
89
90 /* Approximate tan(r) using order 8 polynomial.
91 tan(x) is odd, so polynomial has the form:
92 tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ...
93 Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ...
94 Then compute the approximation by:
95 tan(r) ~= r + r^3 * (C0 + r^2 * P(r)). */
96 float64x2_t r2 = vmulq_f64 (r, r), r4 = vmulq_f64 (r2, r2),
97 r8 = vmulq_f64 (r4, r4);
98 /* Offset coefficients to evaluate from C1 onwards. */
99 float64x2_t p = v_estrin_7_f64 (r2, r4, r8, dat->poly + 1);
100 p = vfmaq_f64 (dat->poly[0], p, r2);
101 p = vfmaq_f64 (r, r2, vmulq_f64 (p, r));
102
103 /* Recombination uses double-angle formula:
104 tan(2x) = 2 * tan(x) / (1 - (tan(x))^2)
105 and reciprocity around pi/2:
106 tan(x) = 1 / (tan(pi/2 - x))
107 to assemble result using change-of-sign and conditional selection of
108 numerator/denominator, dependent on odd/even-ness of q (hence quadrant).
109 */
110 float64x2_t n = vfmaq_f64 (v_f64 (-1), p, p);
111 float64x2_t d = vaddq_f64 (p, p);
112
113 uint64x2_t no_recip = vtstq_u64 (vreinterpretq_u64_s64 (qi), v_u64 (1));
114
115#if !WANT_SIMD_EXCEPT
116 uint64x2_t special = vcageq_f64 (x, dat->range_val);
117 if (__glibc_unlikely (v_any_u64 (special)))
118 return special_case (x);
119#endif
120
121 return vdivq_f64 (vbslq_f64 (no_recip, n, vnegq_f64 (d)),
122 vbslq_f64 (no_recip, d, n));
123}
124

source code of glibc/sysdeps/aarch64/fpu/tan_advsimd.c