1 /*
2 * Double-precision vector cbrt(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 #include "poly_advsimd_f64.h"
12
13 const static struct data
14 {
15 float64x2_t poly[4], one_third, shift;
16 int64x2_t exp_bias;
17 uint64x2_t abs_mask, tiny_bound;
18 uint32x4_t thresh;
19 double table[5];
20 } data = {
21 .shift = V2 (0x1.8p52),
22 .poly = { /* Generated with fpminimax in [0.5, 1]. */
23 V2 (0x1.c14e8ee44767p-2), V2 (0x1.dd2d3f99e4c0ep-1),
24 V2 (-0x1.08e83026b7e74p-1), V2 (0x1.2c74eaa3ba428p-3) },
25 .exp_bias = V2 (1022),
26 .abs_mask = V2(0x7fffffffffffffff),
27 .tiny_bound = V2(0x0010000000000000), /* Smallest normal. */
28 .thresh = V4(0x7fe00000), /* asuint64 (infinity) - tiny_bound. */
29 .one_third = V2(0x1.5555555555555p-2),
30 .table = { /* table[i] = 2^((i - 2) / 3). */
31 0x1.428a2f98d728bp-1, 0x1.965fea53d6e3dp-1, 0x1p0,
32 0x1.428a2f98d728bp0, 0x1.965fea53d6e3dp0 }
33 };
34
35 #define MantissaMask v_u64 (0x000fffffffffffff)
36
37 static float64x2_t NOINLINE VPCS_ATTR
special_case(float64x2_t x,float64x2_t y,uint32x2_t special)38 special_case (float64x2_t x, float64x2_t y, uint32x2_t special)
39 {
40 return v_call_f64 (cbrt, x, y, vmovl_u32 (special));
41 }
42
43 /* Approximation for double-precision vector cbrt(x), using low-order polynomial
44 and two Newton iterations. Greatest observed error is 1.79 ULP. Errors repeat
45 according to the exponent, for instance an error observed for double value
46 m * 2^e will be observed for any input m * 2^(e + 3*i), where i is an
47 integer.
48 __v_cbrt(0x1.fffff403f0bc6p+1) got 0x1.965fe72821e9bp+0
49 want 0x1.965fe72821e99p+0. */
V_NAME_D1(cbrt)50 VPCS_ATTR float64x2_t V_NAME_D1 (cbrt) (float64x2_t x)
51 {
52 const struct data *d = ptr_barrier (&data);
53 uint64x2_t iax = vreinterpretq_u64_f64 (vabsq_f64 (x));
54
55 /* Subnormal, +/-0 and special values. */
56 uint32x2_t special
57 = vcge_u32 (vsubhn_u64 (iax, d->tiny_bound), vget_low_u32 (d->thresh));
58
59 /* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector
60 version of frexp, which gets subnormal values wrong - these have to be
61 special-cased as a result. */
62 float64x2_t m = vbslq_f64 (MantissaMask, x, v_f64 (0.5));
63 int64x2_t exp_bias = d->exp_bias;
64 uint64x2_t ia12 = vshrq_n_u64 (iax, 52);
65 int64x2_t e = vsubq_s64 (vreinterpretq_s64_u64 (ia12), exp_bias);
66
67 /* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point for
68 Newton iterations. */
69 float64x2_t p = v_pairwise_poly_3_f64 (m, vmulq_f64 (m, m), d->poly);
70 float64x2_t one_third = d->one_third;
71 /* Two iterations of Newton's method for iteratively approximating cbrt. */
72 float64x2_t m_by_3 = vmulq_f64 (m, one_third);
73 float64x2_t two_thirds = vaddq_f64 (one_third, one_third);
74 float64x2_t a
75 = vfmaq_f64 (vdivq_f64 (m_by_3, vmulq_f64 (p, p)), two_thirds, p);
76 a = vfmaq_f64 (vdivq_f64 (m_by_3, vmulq_f64 (a, a)), two_thirds, a);
77
78 /* Assemble the result by the following:
79
80 cbrt(x) = cbrt(m) * 2 ^ (e / 3).
81
82 We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is
83 not necessarily a multiple of 3 we lose some information.
84
85 Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q.
86
87 Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which is
88 an integer in [-2, 2], and can be looked up in the table T. Hence the
89 result is assembled as:
90
91 cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign. */
92
93 float64x2_t ef = vcvtq_f64_s64 (e);
94 float64x2_t eb3f = vrndnq_f64 (vmulq_f64 (ef, one_third));
95 int64x2_t em3 = vcvtq_s64_f64 (vfmsq_f64 (ef, eb3f, v_f64 (3)));
96 int64x2_t ey = vcvtq_s64_f64 (eb3f);
97
98 float64x2_t my = (float64x2_t){ d->table[em3[0] + 2], d->table[em3[1] + 2] };
99 my = vmulq_f64 (my, a);
100
101 /* Vector version of ldexp. */
102 float64x2_t y = vreinterpretq_f64_s64 (
103 vshlq_n_s64 (vaddq_s64 (ey, vaddq_s64 (exp_bias, v_s64 (1))), 52));
104 y = vmulq_f64 (y, my);
105
106 if (unlikely (v_any_u32h (special)))
107 return special_case (x, vbslq_f64 (d->abs_mask, y, x), special);
108
109 /* Copy sign. */
110 return vbslq_f64 (d->abs_mask, y, x);
111 }
112
113 PL_TEST_ULP (V_NAME_D1 (cbrt), 1.30)
114 PL_SIG (V, D, 1, cbrt, -10.0, 10.0)
115 PL_TEST_EXPECT_FENV_ALWAYS (V_NAME_D1 (cbrt))
116 PL_TEST_SYM_INTERVAL (V_NAME_D1 (cbrt), 0, inf, 1000000)
117