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
13 const 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
33 static float32x4_t VPCS_ATTR NOINLINE
special_case(float32x4_t x,float32x4_t y,uint16x4_t special)34 special_case (float32x4_t x, float32x4_t y, uint16x4_t special)
35 {
36 return v_call_f32 (cbrtf, x, y, vmovl_u16 (special));
37 }
38
39 static inline float32x4_t
shifted_lookup(const float * table,int32x4_t i)40 shifted_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. */
V_NAME_F1(cbrt)52 VPCS_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
113 PL_SIG (V, F, 1, cbrt, -10.0, 10.0)
114 PL_TEST_ULP (V_NAME_F1 (cbrt), 1.15)
115 PL_TEST_EXPECT_FENV_ALWAYS (V_NAME_F1 (cbrt))
116 PL_TEST_SYM_INTERVAL (V_NAME_F1 (cbrt), 0, inf, 1000000)
117