1 /*
2 * Double-precision vector cosh(x) function.
3 *
4 * Copyright (c) 2022-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "v_math.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11
12 static const struct data
13 {
14 float64x2_t poly[3];
15 float64x2_t inv_ln2, ln2, shift, thres;
16 uint64x2_t index_mask, special_bound;
17 } data = {
18 .poly = { V2 (0x1.fffffffffffd4p-2), V2 (0x1.5555571d6b68cp-3),
19 V2 (0x1.5555576a59599p-5), },
20
21 .inv_ln2 = V2 (0x1.71547652b82fep8), /* N/ln2. */
22 /* -ln2/N. */
23 .ln2 = {-0x1.62e42fefa39efp-9, -0x1.abc9e3b39803f3p-64},
24 .shift = V2 (0x1.8p+52),
25 .thres = V2 (704.0),
26
27 .index_mask = V2 (0xff),
28 /* 0x1.6p9, above which exp overflows. */
29 .special_bound = V2 (0x4086000000000000),
30 };
31
32 static float64x2_t NOINLINE VPCS_ATTR
special_case(float64x2_t x,float64x2_t y,uint64x2_t special)33 special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
34 {
35 return v_call_f64 (cosh, x, y, special);
36 }
37
38 /* Helper for approximating exp(x). Copied from v_exp_tail, with no
39 special-case handling or tail. */
40 static inline float64x2_t
exp_inline(float64x2_t x)41 exp_inline (float64x2_t x)
42 {
43 const struct data *d = ptr_barrier (&data);
44
45 /* n = round(x/(ln2/N)). */
46 float64x2_t z = vfmaq_f64 (d->shift, x, d->inv_ln2);
47 uint64x2_t u = vreinterpretq_u64_f64 (z);
48 float64x2_t n = vsubq_f64 (z, d->shift);
49
50 /* r = x - n*ln2/N. */
51 float64x2_t r = vfmaq_laneq_f64 (x, n, d->ln2, 0);
52 r = vfmaq_laneq_f64 (r, n, d->ln2, 1);
53
54 uint64x2_t e = vshlq_n_u64 (u, 52 - V_EXP_TAIL_TABLE_BITS);
55 uint64x2_t i = vandq_u64 (u, d->index_mask);
56
57 /* y = tail + exp(r) - 1 ~= r + C1 r^2 + C2 r^3 + C3 r^4. */
58 float64x2_t y = vfmaq_f64 (d->poly[1], d->poly[2], r);
59 y = vfmaq_f64 (d->poly[0], y, r);
60 y = vmulq_f64 (vfmaq_f64 (v_f64 (1), y, r), r);
61
62 /* s = 2^(n/N). */
63 u = v_lookup_u64 (__v_exp_tail_data, i);
64 float64x2_t s = vreinterpretq_f64_u64 (vaddq_u64 (u, e));
65
66 return vfmaq_f64 (s, y, s);
67 }
68
69 /* Approximation for vector double-precision cosh(x) using exp_inline.
70 cosh(x) = (exp(x) + exp(-x)) / 2.
71 The greatest observed error is in the scalar fall-back region, so is the
72 same as the scalar routine, 1.93 ULP:
73 _ZGVnN2v_cosh (0x1.628af341989dap+9) got 0x1.fdf28623ef921p+1021
74 want 0x1.fdf28623ef923p+1021.
75
76 The greatest observed error in the non-special region is 1.54 ULP:
77 _ZGVnN2v_cosh (0x1.8e205b6ecacf7p+2) got 0x1.f711dcb0c77afp+7
78 want 0x1.f711dcb0c77b1p+7. */
V_NAME_D1(cosh)79 float64x2_t VPCS_ATTR V_NAME_D1 (cosh) (float64x2_t x)
80 {
81 const struct data *d = ptr_barrier (&data);
82
83 float64x2_t ax = vabsq_f64 (x);
84 uint64x2_t special
85 = vcgtq_u64 (vreinterpretq_u64_f64 (ax), d->special_bound);
86
87 /* Up to the point that exp overflows, we can use it to calculate cosh by
88 exp(|x|) / 2 + 1 / (2 * exp(|x|)). */
89 float64x2_t t = exp_inline (ax);
90 float64x2_t half_t = vmulq_n_f64 (t, 0.5);
91 float64x2_t half_over_t = vdivq_f64 (v_f64 (0.5), t);
92
93 /* Fall back to scalar for any special cases. */
94 if (unlikely (v_any_u64 (special)))
95 return special_case (x, vaddq_f64 (half_t, half_over_t), special);
96
97 return vaddq_f64 (half_t, half_over_t);
98 }
99
100 PL_SIG (V, D, 1, cosh, -10.0, 10.0)
101 PL_TEST_ULP (V_NAME_D1 (cosh), 1.43)
102 PL_TEST_EXPECT_FENV_ALWAYS (V_NAME_D1 (cosh))
103 PL_TEST_SYM_INTERVAL (V_NAME_D1 (cosh), 0, 0x1.6p9, 100000)
104 PL_TEST_SYM_INTERVAL (V_NAME_D1 (cosh), 0x1.6p9, inf, 1000)
105