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
special_case(float64x2_t x)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. */
V_NAME_D1(tan)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