1 /*
2 * Single-precision vector tan(x) function.
3 *
4 * Copyright (c) 2020-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "sv_math.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11
12 static const struct data
13 {
14 float pio2_1, pio2_2, pio2_3, invpio2;
15 float c1, c3, c5;
16 float c0, c2, c4, range_val, shift;
17 } data = {
18 /* Coefficients generated using:
19 poly = fpminimax((tan(sqrt(x))-sqrt(x))/x^(3/2),
20 deg,
21 [|single ...|],
22 [a*a;b*b]);
23 optimize relative error
24 final prec : 23 bits
25 deg : 5
26 a : 0x1p-126 ^ 2
27 b : ((pi) / 0x1p2) ^ 2
28 dirty rel error: 0x1.f7c2e4p-25
29 dirty abs error: 0x1.f7c2ecp-25. */
30 .c0 = 0x1.55555p-2, .c1 = 0x1.11166p-3,
31 .c2 = 0x1.b88a78p-5, .c3 = 0x1.7b5756p-6,
32 .c4 = 0x1.4ef4cep-8, .c5 = 0x1.0e1e74p-7,
33
34 .pio2_1 = 0x1.921fb6p+0f, .pio2_2 = -0x1.777a5cp-25f,
35 .pio2_3 = -0x1.ee59dap-50f, .invpio2 = 0x1.45f306p-1f,
36 .range_val = 0x1p15f, .shift = 0x1.8p+23f
37 };
38
39 static svfloat32_t NOINLINE
special_case(svfloat32_t x,svfloat32_t y,svbool_t cmp)40 special_case (svfloat32_t x, svfloat32_t y, svbool_t cmp)
41 {
42 return sv_call_f32 (tanf, x, y, cmp);
43 }
44
45 /* Fast implementation of SVE tanf.
46 Maximum error is 3.45 ULP:
47 SV_NAME_F1 (tan)(-0x1.e5f0cap+13) got 0x1.ff9856p-1
48 want 0x1.ff9850p-1. */
SV_NAME_F1(tan)49 svfloat32_t SV_NAME_F1 (tan) (svfloat32_t x, const svbool_t pg)
50 {
51 const struct data *d = ptr_barrier (&data);
52
53 /* Determine whether input is too large to perform fast regression. */
54 svbool_t cmp = svacge (pg, x, d->range_val);
55
56 svfloat32_t odd_coeffs = svld1rq (svptrue_b32 (), &d->c1);
57 svfloat32_t pi_vals = svld1rq (svptrue_b32 (), &d->pio2_1);
58
59 /* n = rint(x/(pi/2)). */
60 svfloat32_t q = svmla_lane (sv_f32 (d->shift), x, pi_vals, 3);
61 svfloat32_t n = svsub_x (pg, q, d->shift);
62 /* n is already a signed integer, simply convert it. */
63 svint32_t in = svcvt_s32_x (pg, n);
64 /* Determine if x lives in an interval, where |tan(x)| grows to infinity. */
65 svint32_t alt = svand_x (pg, in, 1);
66 svbool_t pred_alt = svcmpne (pg, alt, 0);
67
68 /* r = x - n * (pi/2) (range reduction into 0 .. pi/4). */
69 svfloat32_t r;
70 r = svmls_lane (x, n, pi_vals, 0);
71 r = svmls_lane (r, n, pi_vals, 1);
72 r = svmls_lane (r, n, pi_vals, 2);
73
74 /* If x lives in an interval, where |tan(x)|
75 - is finite, then use a polynomial approximation of the form
76 tan(r) ~ r + r^3 * P(r^2) = r + r * r^2 * P(r^2).
77 - grows to infinity then use symmetries of tangent and the identity
78 tan(r) = cotan(pi/2 - r) to express tan(x) as 1/tan(-r). Finally, use
79 the same polynomial approximation of tan as above. */
80
81 /* Perform additional reduction if required. */
82 svfloat32_t z = svneg_m (r, pred_alt, r);
83
84 /* Evaluate polynomial approximation of tangent on [-pi/4, pi/4],
85 using Estrin on z^2. */
86 svfloat32_t z2 = svmul_x (pg, z, z);
87 svfloat32_t p01 = svmla_lane (sv_f32 (d->c0), z2, odd_coeffs, 0);
88 svfloat32_t p23 = svmla_lane (sv_f32 (d->c2), z2, odd_coeffs, 1);
89 svfloat32_t p45 = svmla_lane (sv_f32 (d->c4), z2, odd_coeffs, 2);
90
91 svfloat32_t z4 = svmul_x (pg, z2, z2);
92 svfloat32_t p = svmla_x (pg, p01, z4, p23);
93
94 svfloat32_t z8 = svmul_x (pg, z4, z4);
95 p = svmla_x (pg, p, z8, p45);
96
97 svfloat32_t y = svmla_x (pg, z, p, svmul_x (pg, z, z2));
98
99 /* Transform result back, if necessary. */
100 svfloat32_t inv_y = svdivr_x (pg, y, 1.0f);
101
102 /* No need to pass pg to specialcase here since cmp is a strict subset,
103 guaranteed by the cmpge above. */
104 if (unlikely (svptest_any (pg, cmp)))
105 return special_case (x, svsel (pred_alt, inv_y, y), cmp);
106
107 return svsel (pred_alt, inv_y, y);
108 }
109
110 PL_SIG (SV, F, 1, tan, -3.1, 3.1)
111 PL_TEST_ULP (SV_NAME_F1 (tan), 2.96)
112 PL_TEST_INTERVAL (SV_NAME_F1 (tan), -0.0, -0x1p126, 100)
113 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p-149, 0x1p-126, 4000)
114 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p-126, 0x1p-23, 50000)
115 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p-23, 0.7, 50000)
116 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0.7, 1.5, 50000)
117 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 1.5, 100, 50000)
118 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 100, 0x1p17, 50000)
119 PL_TEST_INTERVAL (SV_NAME_F1 (tan), 0x1p17, inf, 50000)
120