1/*
2 * Single-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_f32.h"
12
13const static struct data
14{
15  float32x4_t poly[4], one_third;
16  float table[5];
17} data = {
18  .poly = { /* Very rough approximation of cbrt(x) in [0.5, 1], generated with
19               FPMinimax.  */
20	    V4 (0x1.c14e96p-2), V4 (0x1.dd2d3p-1), V4 (-0x1.08e81ap-1),
21	    V4 (0x1.2c74c2p-3) },
22  .table = { /* table[i] = 2^((i - 2) / 3).  */
23	    0x1.428a3p-1, 0x1.965feap-1, 0x1p0, 0x1.428a3p0, 0x1.965feap0 },
24  .one_third = V4 (0x1.555556p-2f),
25};
26
27#define SignMask v_u32 (0x80000000)
28#define SmallestNormal v_u32 (0x00800000)
29#define Thresh vdup_n_u16 (0x7f00) /* asuint(INFINITY) - SmallestNormal.  */
30#define MantissaMask v_u32 (0x007fffff)
31#define HalfExp v_u32 (0x3f000000)
32
33static float32x4_t VPCS_ATTR NOINLINE
34special_case (float32x4_t x, float32x4_t y, uint16x4_t special)
35{
36  return v_call_f32 (cbrtf, x, y, vmovl_u16 (special));
37}
38
39static inline float32x4_t
40shifted_lookup (const float *table, int32x4_t i)
41{
42  return (float32x4_t){ table[i[0] + 2], table[i[1] + 2], table[i[2] + 2],
43			table[i[3] + 2] };
44}
45
46/* Approximation for vector single-precision cbrt(x) using Newton iteration
47   with initial guess obtained by a low-order polynomial. Greatest error
48   is 1.64 ULP. This is observed for every value where the mantissa is
49   0x1.85a2aa and the exponent is a multiple of 3, for example:
50   _ZGVnN4v_cbrtf(0x1.85a2aap+3) got 0x1.267936p+1
51				want 0x1.267932p+1.  */
52VPCS_ATTR float32x4_t V_NAME_F1 (cbrt) (float32x4_t x)
53{
54  const struct data *d = ptr_barrier (&data);
55  uint32x4_t iax = vreinterpretq_u32_f32 (vabsq_f32 (x));
56
57  /* Subnormal, +/-0 and special values.  */
58  uint16x4_t special = vcge_u16 (vsubhn_u32 (iax, SmallestNormal), Thresh);
59
60  /* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector
61     version of frexpf, which gets subnormal values wrong - these have to be
62     special-cased as a result.  */
63  float32x4_t m = vbslq_f32 (MantissaMask, x, v_f32 (0.5));
64  int32x4_t e
65      = vsubq_s32 (vreinterpretq_s32_u32 (vshrq_n_u32 (iax, 23)), v_s32 (126));
66
67  /* p is a rough approximation for cbrt(m) in [0.5, 1.0]. The better this is,
68     the less accurate the next stage of the algorithm needs to be. An order-4
69     polynomial is enough for one Newton iteration.  */
70  float32x4_t p = v_pairwise_poly_3_f32 (m, vmulq_f32 (m, m), d->poly);
71
72  float32x4_t one_third = d->one_third;
73  float32x4_t two_thirds = vaddq_f32 (one_third, one_third);
74
75  /* One iteration of Newton's method for iteratively approximating cbrt.  */
76  float32x4_t m_by_3 = vmulq_f32 (m, one_third);
77  float32x4_t a
78      = vfmaq_f32 (vdivq_f32 (m_by_3, vmulq_f32 (p, p)), two_thirds, p);
79
80  /* Assemble the result by the following:
81
82     cbrt(x) = cbrt(m) * 2 ^ (e / 3).
83
84     We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is
85     not necessarily a multiple of 3 we lose some information.
86
87     Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q.
88
89     Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which
90     is an integer in [-2, 2], and can be looked up in the table T. Hence the
91     result is assembled as:
92
93     cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign.  */
94  float32x4_t ef = vmulq_f32 (vcvtq_f32_s32 (e), one_third);
95  int32x4_t ey = vcvtq_s32_f32 (ef);
96  int32x4_t em3 = vsubq_s32 (e, vmulq_s32 (ey, v_s32 (3)));
97
98  float32x4_t my = shifted_lookup (d->table, em3);
99  my = vmulq_f32 (my, a);
100
101  /* Vector version of ldexpf.  */
102  float32x4_t y
103      = vreinterpretq_f32_s32 (vshlq_n_s32 (vaddq_s32 (ey, v_s32 (127)), 23));
104  y = vmulq_f32 (y, my);
105
106  if (unlikely (v_any_u16h (special)))
107    return special_case (x, vbslq_f32 (SignMask, x, y), special);
108
109  /* Copy sign.  */
110  return vbslq_f32 (SignMask, x, y);
111}
112
113PL_SIG (V, F, 1, cbrt, -10.0, 10.0)
114PL_TEST_ULP (V_NAME_F1 (cbrt), 1.15)
115PL_TEST_EXPECT_FENV_ALWAYS (V_NAME_F1 (cbrt))
116PL_TEST_SYM_INTERVAL (V_NAME_F1 (cbrt), 0, inf, 1000000)
117