1*5a02ffc3SAndrew Turner /*
2*5a02ffc3SAndrew Turner * Double-precision SVE cosh(x) function.
3*5a02ffc3SAndrew Turner *
4*5a02ffc3SAndrew Turner * Copyright (c) 2023, Arm Limited.
5*5a02ffc3SAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*5a02ffc3SAndrew Turner */
7*5a02ffc3SAndrew Turner
8*5a02ffc3SAndrew Turner #include "sv_math.h"
9*5a02ffc3SAndrew Turner #include "pl_sig.h"
10*5a02ffc3SAndrew Turner #include "pl_test.h"
11*5a02ffc3SAndrew Turner
12*5a02ffc3SAndrew Turner static const struct data
13*5a02ffc3SAndrew Turner {
14*5a02ffc3SAndrew Turner float64_t poly[3];
15*5a02ffc3SAndrew Turner float64_t inv_ln2, ln2_hi, ln2_lo, shift, thres;
16*5a02ffc3SAndrew Turner uint64_t index_mask, special_bound;
17*5a02ffc3SAndrew Turner } data = {
18*5a02ffc3SAndrew Turner .poly = { 0x1.fffffffffffd4p-2, 0x1.5555571d6b68cp-3,
19*5a02ffc3SAndrew Turner 0x1.5555576a59599p-5, },
20*5a02ffc3SAndrew Turner
21*5a02ffc3SAndrew Turner .inv_ln2 = 0x1.71547652b82fep8, /* N/ln2. */
22*5a02ffc3SAndrew Turner /* -ln2/N. */
23*5a02ffc3SAndrew Turner .ln2_hi = -0x1.62e42fefa39efp-9,
24*5a02ffc3SAndrew Turner .ln2_lo = -0x1.abc9e3b39803f3p-64,
25*5a02ffc3SAndrew Turner .shift = 0x1.8p+52,
26*5a02ffc3SAndrew Turner .thres = 704.0,
27*5a02ffc3SAndrew Turner
28*5a02ffc3SAndrew Turner .index_mask = 0xff,
29*5a02ffc3SAndrew Turner /* 0x1.6p9, above which exp overflows. */
30*5a02ffc3SAndrew Turner .special_bound = 0x4086000000000000,
31*5a02ffc3SAndrew Turner };
32*5a02ffc3SAndrew Turner
33*5a02ffc3SAndrew Turner static svfloat64_t NOINLINE
special_case(svfloat64_t x,svfloat64_t y,svbool_t special)34*5a02ffc3SAndrew Turner special_case (svfloat64_t x, svfloat64_t y, svbool_t special)
35*5a02ffc3SAndrew Turner {
36*5a02ffc3SAndrew Turner return sv_call_f64 (cosh, x, y, special);
37*5a02ffc3SAndrew Turner }
38*5a02ffc3SAndrew Turner
39*5a02ffc3SAndrew Turner /* Helper for approximating exp(x). Copied from sv_exp_tail, with no
40*5a02ffc3SAndrew Turner special-case handling or tail. */
41*5a02ffc3SAndrew Turner static inline svfloat64_t
exp_inline(svfloat64_t x,const svbool_t pg,const struct data * d)42*5a02ffc3SAndrew Turner exp_inline (svfloat64_t x, const svbool_t pg, const struct data *d)
43*5a02ffc3SAndrew Turner {
44*5a02ffc3SAndrew Turner /* Calculate exp(x). */
45*5a02ffc3SAndrew Turner svfloat64_t z = svmla_x (pg, sv_f64 (d->shift), x, d->inv_ln2);
46*5a02ffc3SAndrew Turner svfloat64_t n = svsub_x (pg, z, d->shift);
47*5a02ffc3SAndrew Turner
48*5a02ffc3SAndrew Turner svfloat64_t r = svmla_x (pg, x, n, d->ln2_hi);
49*5a02ffc3SAndrew Turner r = svmla_x (pg, r, n, d->ln2_lo);
50*5a02ffc3SAndrew Turner
51*5a02ffc3SAndrew Turner svuint64_t u = svreinterpret_u64 (z);
52*5a02ffc3SAndrew Turner svuint64_t e = svlsl_x (pg, u, 52 - V_EXP_TAIL_TABLE_BITS);
53*5a02ffc3SAndrew Turner svuint64_t i = svand_x (pg, u, d->index_mask);
54*5a02ffc3SAndrew Turner
55*5a02ffc3SAndrew Turner svfloat64_t y = svmla_x (pg, sv_f64 (d->poly[1]), r, d->poly[2]);
56*5a02ffc3SAndrew Turner y = svmla_x (pg, sv_f64 (d->poly[0]), r, y);
57*5a02ffc3SAndrew Turner y = svmla_x (pg, sv_f64 (1.0), r, y);
58*5a02ffc3SAndrew Turner y = svmul_x (pg, r, y);
59*5a02ffc3SAndrew Turner
60*5a02ffc3SAndrew Turner /* s = 2^(n/N). */
61*5a02ffc3SAndrew Turner u = svld1_gather_index (pg, __v_exp_tail_data, i);
62*5a02ffc3SAndrew Turner svfloat64_t s = svreinterpret_f64 (svadd_x (pg, u, e));
63*5a02ffc3SAndrew Turner
64*5a02ffc3SAndrew Turner return svmla_x (pg, s, s, y);
65*5a02ffc3SAndrew Turner }
66*5a02ffc3SAndrew Turner
67*5a02ffc3SAndrew Turner /* Approximation for SVE double-precision cosh(x) using exp_inline.
68*5a02ffc3SAndrew Turner cosh(x) = (exp(x) + exp(-x)) / 2.
69*5a02ffc3SAndrew Turner The greatest observed error is in the scalar fall-back region, so is the
70*5a02ffc3SAndrew Turner same as the scalar routine, 1.93 ULP:
71*5a02ffc3SAndrew Turner _ZGVsMxv_cosh (0x1.628ad45039d2fp+9) got 0x1.fd774e958236dp+1021
72*5a02ffc3SAndrew Turner want 0x1.fd774e958236fp+1021.
73*5a02ffc3SAndrew Turner
74*5a02ffc3SAndrew Turner The greatest observed error in the non-special region is 1.54 ULP:
75*5a02ffc3SAndrew Turner _ZGVsMxv_cosh (0x1.ba5651dd4486bp+2) got 0x1.f5e2bb8d5c98fp+8
76*5a02ffc3SAndrew Turner want 0x1.f5e2bb8d5c991p+8. */
SV_NAME_D1(cosh)77*5a02ffc3SAndrew Turner svfloat64_t SV_NAME_D1 (cosh) (svfloat64_t x, const svbool_t pg)
78*5a02ffc3SAndrew Turner {
79*5a02ffc3SAndrew Turner const struct data *d = ptr_barrier (&data);
80*5a02ffc3SAndrew Turner
81*5a02ffc3SAndrew Turner svfloat64_t ax = svabs_x (pg, x);
82*5a02ffc3SAndrew Turner svbool_t special = svcmpgt (pg, svreinterpret_u64 (ax), d->special_bound);
83*5a02ffc3SAndrew Turner
84*5a02ffc3SAndrew Turner /* Up to the point that exp overflows, we can use it to calculate cosh by
85*5a02ffc3SAndrew Turner exp(|x|) / 2 + 1 / (2 * exp(|x|)). */
86*5a02ffc3SAndrew Turner svfloat64_t t = exp_inline (ax, pg, d);
87*5a02ffc3SAndrew Turner svfloat64_t half_t = svmul_x (pg, t, 0.5);
88*5a02ffc3SAndrew Turner svfloat64_t half_over_t = svdivr_x (pg, t, 0.5);
89*5a02ffc3SAndrew Turner
90*5a02ffc3SAndrew Turner /* Fall back to scalar for any special cases. */
91*5a02ffc3SAndrew Turner if (unlikely (svptest_any (pg, special)))
92*5a02ffc3SAndrew Turner return special_case (x, svadd_x (pg, half_t, half_over_t), special);
93*5a02ffc3SAndrew Turner
94*5a02ffc3SAndrew Turner return svadd_x (pg, half_t, half_over_t);
95*5a02ffc3SAndrew Turner }
96*5a02ffc3SAndrew Turner
97*5a02ffc3SAndrew Turner PL_SIG (SV, D, 1, cosh, -10.0, 10.0)
98*5a02ffc3SAndrew Turner PL_TEST_ULP (SV_NAME_D1 (cosh), 1.43)
99*5a02ffc3SAndrew Turner PL_TEST_SYM_INTERVAL (SV_NAME_D1 (cosh), 0, 0x1.6p9, 100000)
100*5a02ffc3SAndrew Turner PL_TEST_SYM_INTERVAL (SV_NAME_D1 (cosh), 0x1.6p9, inf, 1000)
101