1 /*
2 * Double-precision SVE tan(x) function.
3 *
4 * Copyright (c) 2023-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "sv_math.h"
9 #include "test_sig.h"
10 #include "test_defs.h"
11
12 static const struct data
13 {
14 double c2, c4, c6, c8;
15 double poly_1357[4];
16 double c0, inv_half_pi;
17 double half_pi_hi, half_pi_lo, range_val;
18 } data = {
19 /* Polynomial generated with FPMinimax. */
20 .c2 = 0x1.ba1ba1bb46414p-5,
21 .c4 = 0x1.226e5e5ecdfa3p-7,
22 .c6 = 0x1.7ea75d05b583ep-10,
23 .c8 = 0x1.4e4fd14147622p-12,
24 .poly_1357 = { 0x1.1111111110a63p-3, 0x1.664f47e5b5445p-6,
25 0x1.d6c7ddbf87047p-9, 0x1.289f22964a03cp-11 },
26 .c0 = 0x1.5555555555556p-2,
27 .inv_half_pi = 0x1.45f306dc9c883p-1,
28 .half_pi_hi = 0x1.921fb54442d18p0,
29 .half_pi_lo = 0x1.1a62633145c07p-54,
30 .range_val = 0x1p23,
31 };
32
33 static svfloat64_t NOINLINE
special_case(svfloat64_t x,svfloat64_t p,svfloat64_t q,svbool_t pg,svbool_t special)34 special_case (svfloat64_t x, svfloat64_t p, svfloat64_t q, svbool_t pg,
35 svbool_t special)
36 {
37 svbool_t use_recip = svcmpeq (
38 pg, svand_x (pg, svreinterpret_u64 (svcvt_s64_x (pg, q)), 1), 0);
39
40 svfloat64_t n = svmad_x (pg, p, p, -1);
41 svfloat64_t d = svmul_x (svptrue_b64 (), p, 2);
42 svfloat64_t swap = n;
43 n = svneg_m (n, use_recip, d);
44 d = svsel (use_recip, swap, d);
45 svfloat64_t y = svdiv_x (svnot_z (pg, special), n, d);
46 return sv_call_f64 (tan, x, y, special);
47 }
48
49 /* Vector approximation for double-precision tan.
50 Maximum measured error is 3.48 ULP:
51 _ZGVsMxv_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37
52 want -0x1.f6ccd8ecf7deap+37. */
SV_NAME_D1(tan)53 svfloat64_t SV_NAME_D1 (tan) (svfloat64_t x, svbool_t pg)
54 {
55 const struct data *dat = ptr_barrier (&data);
56 svfloat64_t half_pi_c0 = svld1rq (svptrue_b64 (), &dat->c0);
57 /* q = nearest integer to 2 * x / pi. */
58 svfloat64_t q = svmul_lane (x, half_pi_c0, 1);
59 q = svrinta_x (pg, q);
60
61 /* Use q to reduce x to r in [-pi/4, pi/4], by:
62 r = x - q * pi/2, in extended precision. */
63 svfloat64_t r = x;
64 svfloat64_t half_pi = svld1rq (svptrue_b64 (), &dat->half_pi_hi);
65 r = svmls_lane (r, q, half_pi, 0);
66 r = svmls_lane (r, q, half_pi, 1);
67 /* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle
68 formula. */
69 r = svmul_x (svptrue_b64 (), r, 0.5);
70
71 /* Approximate tan(r) using order 8 polynomial.
72 tan(x) is odd, so polynomial has the form:
73 tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ...
74 Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ...
75 Then compute the approximation by:
76 tan(r) ~= r + r^3 * (C0 + r^2 * P(r)). */
77
78 svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r);
79 svfloat64_t r4 = svmul_x (svptrue_b64 (), r2, r2);
80 svfloat64_t r8 = svmul_x (svptrue_b64 (), r4, r4);
81 /* Use offset version coeff array by 1 to evaluate from C1 onwards. */
82 svfloat64_t C_24 = svld1rq (svptrue_b64 (), &dat->c2);
83 svfloat64_t C_68 = svld1rq (svptrue_b64 (), &dat->c6);
84
85 /* Use offset version coeff array by 1 to evaluate from C1 onwards. */
86 svfloat64_t p01 = svmla_lane (sv_f64 (dat->poly_1357[0]), r2, C_24, 0);
87 svfloat64_t p23 = svmla_lane_f64 (sv_f64 (dat->poly_1357[1]), r2, C_24, 1);
88 svfloat64_t p03 = svmla_x (pg, p01, p23, r4);
89
90 svfloat64_t p45 = svmla_lane (sv_f64 (dat->poly_1357[2]), r2, C_68, 0);
91 svfloat64_t p67 = svmla_lane (sv_f64 (dat->poly_1357[3]), r2, C_68, 1);
92 svfloat64_t p47 = svmla_x (pg, p45, p67, r4);
93
94 svfloat64_t p = svmla_x (pg, p03, p47, r8);
95
96 svfloat64_t z = svmul_x (svptrue_b64 (), p, r);
97 z = svmul_x (svptrue_b64 (), r2, z);
98 z = svmla_lane (z, r, half_pi_c0, 0);
99 p = svmla_x (pg, r, r2, z);
100
101 /* Recombination uses double-angle formula:
102 tan(2x) = 2 * tan(x) / (1 - (tan(x))^2)
103 and reciprocity around pi/2:
104 tan(x) = 1 / (tan(pi/2 - x))
105 to assemble result using change-of-sign and conditional selection of
106 numerator/denominator dependent on odd/even-ness of q (quadrant). */
107
108 /* Invert condition to catch NaNs and Infs as well as large values. */
109 svbool_t special = svnot_z (pg, svaclt (pg, x, dat->range_val));
110
111 if (unlikely (svptest_any (pg, special)))
112 {
113 return special_case (x, p, q, pg, special);
114 }
115 svbool_t use_recip = svcmpeq (
116 pg, svand_x (pg, svreinterpret_u64 (svcvt_s64_x (pg, q)), 1), 0);
117
118 svfloat64_t n = svmad_x (pg, p, p, -1);
119 svfloat64_t d = svmul_x (svptrue_b64 (), p, 2);
120 svfloat64_t swap = n;
121 n = svneg_m (n, use_recip, d);
122 d = svsel (use_recip, swap, d);
123 return svdiv_x (pg, n, d);
124 }
125
126 TEST_SIG (SV, D, 1, tan, -3.1, 3.1)
127 TEST_ULP (SV_NAME_D1 (tan), 2.99)
128 TEST_DISABLE_FENV (SV_NAME_D1 (tan))
129 TEST_SYM_INTERVAL (SV_NAME_D1 (tan), 0, 0x1p23, 500000)
130 TEST_SYM_INTERVAL (SV_NAME_D1 (tan), 0x1p23, inf, 5000)
131 CLOSE_SVE_ATTR
132