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

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